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Operations with Variable Exponents

43 questions

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Question 21Question →

A student is asked to simplify the rational algebraic expression shown below.

(3a2b−3)23a−1b−4 \frac{(3a^2b^{-3})^2}{3a^{-1}b^{-4}}

What is the fully simplified form of this expression with positive exponents?

Show answer & explanation

Answer: 3a5b2\frac{3a^5}{b^2}

Answer

3a5b2\frac{3a^5}{b^2}
To simplify the expression, first simplify the numerator by squaring every factor inside the parentheses: (3)2⋅(a2)2⋅(b−3)2(3)^2 \cdot (a^2)^2 \cdot (b^{-3})^2, which yields 9a4b−69a^4b^{-6}. Next, divide this result by the denominator 3a−1b−43a^{-1}b^{-4}. Divide the coefficients: 9÷3=39 \div 3 = 3. Then, apply the quotient rule by subtracting the exponents of like bases: for aa, 4−(−1)=54 - (-1) = 5; for bb, −6−(−4)=−2-6 - (-4) = -2. This gives 3a5b−23a^5b^{-2}. Finally, rewrite the expression with positive exponents by moving b−2b^{-2} to the denominator, resulting in 3a5b2\frac{3a^5}{b^2}.

Step-by-Step Solution

1
Apply the power of a product and power of a power rules to the numerator: (3a2b−3)2(3a^2b^{-3})^2.
9a4b−69a^4b^{-6}
The exponent outside the parentheses applies to the coefficient and is multiplied by each variable's exponent.
2
Substitute the expanded numerator back into the fraction.
9a4b−63a−1b−4\frac{9a^4b^{-6}}{3a^{-1}b^{-4}}
This sets up the expression for the quotient rule.
3
Divide the numerical coefficients.
9÷3=39 \div 3 = 3
Coefficients are divided normally.
4
Apply the quotient rule by subtracting the denominator's exponents from the numerator's exponents for bases aa and bb.
a4−(−1)=a5a^{4 - (-1)} = a^5 and b−6−(−4)=b−2b^{-6 - (-4)} = b^{-2}
When dividing like bases, exponents are subtracted. Care must be taken with double negative signs.
5
Combine the results and convert any negative exponents to positive exponents by moving them to the denominator.
3a5b−2=3a5b23a^5b^{-2} = \frac{3a^5}{b^2}
Standard mathematical notation requires final answers to be expressed using positive exponents.

Key Concept

Applying exponent rules (power, product, and quotient) with numerical coefficients and negative integers.
Question 22Question →

Match each unsimplified algebraic expression on the left with its equivalent fully simplified form on the right. Assume all variables represent non-zero real numbers.

Click a left item, then click its matching right item

Items

(3a4b2)(−4a2b−5)(3a^4b^2)(-4a^2b^{-5})
16a3b68a7b2\frac{16a^3b^6}{8a^7b^2}
(2a3b5)4(2a^3b^5)^4
(5a−2b0)−2(5a^{-2}b^0)^{-2}

Matches

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Answer

The product rule matches to -12a^6/b^3, the quotient rule matches to 2b^4/a^4, the power of a product rule matches to 16a^12b^20, and the negative exponent simplification matches to a^4/25.
Matching each pair correctly requires correctly identifying and applying four distinct fundamental laws of exponents: the product rule (adding exponents), the quotient rule (subtracting exponents), the power of a product rule (multiplying exponents), and the rules for evaluating zero and negative exponents.

Step-by-Step Solution

1
Simplify (3a4b2)(−4a2b−5)(3a^4b^2)(-4a^2b^{-5}) using the product rule for exponents.
−12a6b−3-12a^6b^{-3}, which simplifies to −12a6b3\frac{-12a^6}{b^3}.
When multiplying expressions with the same base, add their exponents and multiply their numerical coefficients.
2
Simplify 16a3b68a7b2\frac{16a^3b^6}{8a^7b^2} using the quotient rule for exponents.
2a−4b42a^{-4}b^4, which simplifies to 2b4a4\frac{2b^4}{a^4}.
When dividing expressions with the same base, subtract the denominator's exponent from the numerator's exponent and divide the numerical coefficients.
3
Simplify (2a3b5)4(2a^3b^5)^4 using the power of a product rule.
16a12b2016a^{12}b^{20}.
When raising a product to a power, raise the numerical coefficient to that power and multiply each variable's exponent by the outer power.
4
Simplify (5a−2b0)−2(5a^{-2}b^0)^{-2} using the zero and negative exponent rules.
First, b0=1b^0 = 1. Then evaluate 5−2a45^{-2}a^4, which simplifies to a425\frac{a^4}{25}.
Any non-zero term raised to the zero power equals 1. Negative exponents indicate the reciprocal of the base.

Key Concept

Operations with Variable Exponents
Question 23Question →

Calculate the product of the two algebraic terms below:

−4m3×3m4 -4m^3 \times 3m^4

What is the correct result of this multiplication?

Show answer & explanation

Answer: −12m7-12m^7

Answer

The correct answer is −12m7-12m^7.
To multiply these terms, first multiply the numerical coefficients: −4×3=−12-4 \times 3 = -12. Then, apply the product rule for exponents by adding the exponents of the common base mm: 3+4=73 + 4 = 7. Combining these yields −12m7-12m^7.

Step-by-Step Solution

1
Multiply the numerical coefficients.
−4×3=−12-4 \times 3 = -12
When multiplying algebraic terms, the constant factors are multiplied together using standard integer rules.
2
Apply the product rule for exponents to the variables.
m3×m4=m3+4=m7m^3 \times m^4 = m^{3+4} = m^7
The product rule states that when multiplying terms with the same base, their exponents should be added.
3
Combine the coefficient and the variable part.
−12m7-12m^7
Putting the results from the previous steps together gives the final simplified expression.

Key Concept

Product Rule for Exponents and Multiplying Monomials
Estimated Time:45s
Question 24Question →

During a physics calculation, a student models a system's energy output using the following expression, where xx and yy are non-zero variables:

(2x−2y3)34x−4y5 \frac{(2x^{-2}y^3)^3}{4x^{-4}y^5}

What is the fully simplified form of this expression using only positive exponents?

Show answer & explanation

Answer: 2y4x2\frac{2y^4}{x^2}

Answer

2y4x2\frac{2y^4}{x^2}
Applying the power of a product rule to the numerator yields 8x−6y98x^{-6}y^9. Dividing this by the denominator 4x−4y54x^{-4}y^5 involves dividing the coefficients (8/4=28/4 = 2) and subtracting the exponents of like bases. For xx, −6−(−4)=−2-6 - (-4) = -2, and for yy, 9−5=49 - 5 = 4. The resulting expression is 2x−2y42x^{-2}y^4. To represent this with positive exponents, the x−2x^{-2} term is moved to the denominator, giving 2y4x2\frac{2y^4}{x^2}.

Step-by-Step Solution

1
Apply the power of a product rule to the numerator.
(2x−2y3)3=23⋅(x−2)3⋅(y3)3=8x−6y9(2x^{-2}y^3)^3 = 2^3 \cdot (x^{-2})^3 \cdot (y^3)^3 = 8x^{-6}y^9
The exponent outside the parentheses applies to every individual factor inside.
2
Divide the numerical coefficients.
8÷4=28 \div 4 = 2
Numerical coefficients are handled separately from variables with exponents.
3
Apply the quotient of powers rule to the variables.
x−6−(−4)=x−2x^{-6 - (-4)} = x^{-2} and y9−5=y4y^{9 - 5} = y^4
When dividing like bases, you subtract the denominator's exponent from the numerator's exponent.
4
Combine the results and convert the negative exponent to a positive exponent.
2x−2y4=2y4x22x^{-2}y^4 = \frac{2y^4}{x^2}
Moving a term with a negative exponent to the denominator changes its exponent to positive.

Key Concept

Simplifying rational algebraic expressions using the power of a product rule, the quotient of powers rule, and negative exponents.
Question 25Question →

Simplify the following algebraic expression, assuming all variables are non-zero:

(3m2n4)23m5n3 \frac{(3m^2n^4)^2}{3m^5n^3}

Which of the following represents the fully simplified form?

Show answer & explanation

Answer: 3n5m\frac{3n^5}{m}

Answer

The fully simplified expression is 3n5m\frac{3n^5}{m}.
The correct expression is derived by properly applying the power rules. First, squaring the numerator (3m2n4)2(3m^2n^4)^2 yields 9m4n89m^4n^8. Then, dividing this by the denominator 3m5n33m^5n^3 requires dividing the coefficients (9/3=39/3 = 3) and subtracting the exponents of corresponding variables (4−5=−14-5 = -1 for mm, and 8−3=58-3 = 5 for nn). Finally, rewriting 3m−1n53m^{-1}n^5 with only positive exponents gives 3n5m\frac{3n^5}{m}.

Step-by-Step Solution

1
Simplify the numerator using the power of a product rule.
(3m2n4)2=32⋅(m2)2⋅(n4)2=9m4n8(3m^2n^4)^2 = 3^2 \cdot (m^2)^2 \cdot (n^4)^2 = 9m^4n^8
When a product is raised to a power, each factor inside the parentheses must be raised to that power.
2
Substitute the simplified numerator back into the fraction.
9m4n83m5n3\frac{9m^4n^8}{3m^5n^3}
This sets up the expression for applying the quotient rules.
3
Divide the numerical coefficients.
93=3\frac{9}{3} = 3
Constants are simplified through standard division.
4
Apply the quotient rule for exponents to the variables mm and nn.
m4−5=m−1m^{4-5} = m^{-1} and n8−3=n5n^{8-3} = n^5
When dividing terms with the same base, subtract the exponent in the denominator from the exponent in the numerator.
5
Combine the results and rewrite without negative exponents.
3m−1n5=3n5m3m^{-1}n^5 = \frac{3n^5}{m}
A negative exponent indicates the reciprocal of the base, moving the term to the denominator.

Key Concept

Exponent rules including power of a power, power of a product, and quotient of powers.
Question 26Question →
If the equation (3x4yk)2x3y5=9x5y11\frac{(3x^4 y^k)^2}{x^3 y^5} = 9x^5 y^{11} is true for all non-zero values of xx and yy, what is the value of kk?
Show answer & explanation

Answer: 8; k=8; k = 8

Answer

8
By applying the power rule to expand the numerator, the expression becomes 9x8y2k9x^8 y^{2k}. Using the quotient rule to divide these terms by the denominator x3y5x^3 y^5, the expression simplifies to 9x5y2k−59x^5 y^{2k-5}. Equating this final simplified form to the given result 9x5y119x^5 y^{11}, we can isolate the exponent of yy to set up the equation 2k−5=112k - 5 = 11. Solving this equation gives k=8k = 8.

Step-by-Step Solution

1
Apply the power of a product and power of a power rules to the numerator.
(3x4yk)2=32(x4)2(yk)2=9x8y2k(3x^4 y^k)^2 = 3^2 (x^4)^2 (y^k)^2 = 9x^8 y^{2k}
When a product is raised to a power, each factor inside the parentheses must be individually raised to that power by multiplying the exponents.
2
Substitute the expanded numerator back into the fraction and apply the quotient rule.
9x8y2kx3y5=9x8−3y2k−5=9x5y2k−5\frac{9x^8 y^{2k}}{x^3 y^5} = 9 x^{8-3} y^{2k-5} = 9x^5 y^{2k-5}
When dividing algebraic terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.
3
Equate the simplified exponent of yy to the exponent of yy on the right side of the original equation.
2k−5=112k - 5 = 11
Because the expressions are equivalent and their coefficients and xx-terms already match, their yy-exponents must be strictly equal.
4
Solve the resulting linear equation for kk.
2k=16→k=82k = 16 \rightarrow k = 8
Adding 5 to both sides and then dividing by 2 isolates the variable kk.

Key Concept

Operations with Variable Exponents
Estimated Time:1m 30s
Question 27Question →

In a structural modeling calculation, an engineer must simplify a given formula before evaluating it. Which of the following expressions is mathematically equivalent to the algebraic fraction below?

16a5b2(2ab−1)3\frac{16a^5b^2}{(2ab^{-1})^3}
Show answer & explanation

Answer: 2a2b52a^2b^5

Answer

The mathematically equivalent expression is 2a2b52a^2b^5.
The correct answer is derived by first applying the power rule to the entire denominator, yielding 8a3b−38a^3b^{-3}. Next, dividing the numerical coefficients (16÷8=216 \div 8 = 2) and applying the quotient rule for exponents (subtracting the denominator's exponents from the numerator's exponents) results in a5−3=a2a^{5-3} = a^2 and b2−(−3)=b2+3=b5b^{2-(-3)} = b^{2+3} = b^5. Combining these distinct parts yields 2a2b52a^2b^5.

Step-by-Step Solution

1
Apply the power of a product rule to the denominator.
(2ab−1)3=23⋅a3⋅(b−1)3(2ab^{-1})^3 = 2^3 \cdot a^3 \cdot (b^{-1})^3
An exponent outside parentheses must be applied to every distinct factor inside the parentheses.
2
Simplify the factors in the denominator.
8a3b−38a^3b^{-3}
23=82^3 = 8, and for a power of a power (b−1)3(b^{-1})^3, the exponents are multiplied (−1×3=−3-1 \times 3 = -3).
3
Rewrite the fraction using the expanded denominator.
16a5b28a3b−3\frac{16a^5b^2}{8a^3b^{-3}}
This sets up the expression for application of the quotient rule.
4
Divide the numerical coefficients.
16÷8=216 \div 8 = 2
Constants are separated from variables and divided normally.
5
Apply the quotient rule to the variables with the base aa.
a5−3=a2a^{5-3} = a^2
When dividing like bases, subtract the exponent of the denominator from the exponent of the numerator.
6
Apply the quotient rule to the variables with the base bb.
b2−(−3)=b2+3=b5b^{2 - (-3)} = b^{2 + 3} = b^5
Subtracting a negative exponent is equivalent to adding a positive number.
7
Combine the simplified coefficient and variables.
2a2b52a^2b^5
Bringing the distinct evaluated parts back together provides the final simplified term.

Key Concept

To simplify a rational expression involving exponents, first use the power rule to eliminate grouping symbols, ensuring numerical coefficients are raised to the power. Then, divide the coefficients and use the quotient rule to subtract the denominator's exponents from the numerator's exponents for matching bases.
Estimated Time:1m 30s
Question 28Question →

Match each unsimplified algebraic expression on the left with its fully simplified equivalent form on the right. Assume pp and qq represent non-zero numbers.

Click a left item, then click its matching right item

Items

(4p5q−2)(2p−1q6) (4p^5 q^{-2})(2p^{-1} q^6)
15p7q23p2q−3 \frac{15p^7 q^2}{3p^2 q^{-3}}
(3p3q2)2 (3p^3 q^2)^2
(2p2q3)34p4q7 \frac{(2p^2 q^3)^3}{4p^4 q^7}

Matches

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Answer

The correct matches are: (4p^5 q^-2)(2p^-1 q^6) matches 8p^4 q^4; the fraction 15p^7 q^2 over 3p^2 q^-3 matches 5p^5 q^5; (3p^3 q^2)^2 matches 9p^6 q^4; and the fraction (2p^2 q^3)^3 over 4p^4 q^7 matches 2p^2 q^2.
Each expression simplifies to exactly one of the given target forms by consistently applying the laws of exponents (product, quotient, and power rules) and handling numerical coefficients accurately.

Step-by-Step Solution

1
Simplify the first expression using the product rule for exponents.
The coefficients multiply to 8. Adding the exponents for p gives 5 + (-1) = 4. Adding the exponents for q gives -2 + 6 = 4. The result is 8p^4 q^4.
When multiplying terms with the same base, you add their exponents.
2
Simplify the second expression using the quotient rule for exponents.
The coefficients divide to 5. Subtracting the p exponents gives 7 - 2 = 5. Subtracting the q exponents gives 2 - (-3) = 5. The result is 5p^5 q^5.
When dividing terms with the same base, you subtract the denominator's exponent from the numerator's exponent.
3
Simplify the third expression using the power of a product rule.
The coefficient squared is 3^2 = 9. Multiplying the p exponent by 2 gives 3 * 2 = 6. Multiplying the q exponent by 2 gives 2 * 2 = 4. The result is 9p^6 q^4.
When raising a power to a power, multiply the exponents. Remember to also apply the power to the numerical coefficient.
4
Simplify the fourth expression by applying the power rule first, then the quotient rule.
The numerator becomes 8p^6 q^9. Dividing this by 4p^4 q^7 gives a coefficient of 2, p exponent of 6 - 4 = 2, and q exponent of 9 - 7 = 2. The result is 2p^2 q^2.
Order of operations requires handling the exponent on the entire numerator before performing division.

Key Concept

Applying the product, quotient, and power rules of exponents to simplify algebraic expressions.
Question 29Question →

Match each unsimplified algebraic expression on the left with its equivalent, fully simplified form on the right. Assume all variables represent non-zero real numbers.

Click a left item, then click its matching right item

Items

(3a4b2)(4a2b)(3a^4b^2)(4a^2b)
24a7b36a3b6\frac{24a^7b^3}{6a^3b^6}
(2a3b−2)4(2a^3b^{-2})^4
(5a0b3)−2(5a^0b^3)^{-2}

Matches

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Answer

The expressions match as follows: (3a^4b^2)(4a^2b) is equivalent to 12a^6b^3. The quotient (24a^7b^3)/(6a^3b^6) simplifies to 4a^4b^-3. The power (2a^3b^-2)^4 expands to 16a^12b^-8. Finally, (5a^0b^3)^-2 simplifies to 1/(25b^6).
Each expression simplifies directly using the fundamental rules of exponents: product rule (add exponents), quotient rule (subtract exponents), power rule (multiply exponents), and zero/negative exponent rules.

Step-by-Step Solution

1
Simplify the first expression by multiplying the coefficients and applying the product rule for exponents.
3 * 4 = 12, a^(4+2) = a^6, b^(2+1) = b^3. Combined: 12a^6b^3
When multiplying terms with the same base, add their exponents.
2
Simplify the second expression by dividing the coefficients and applying the quotient rule for exponents.
24 / 6 = 4, a^(7-3) = a^4, b^(3-6) = b^-3. Combined: 4a^4b^-3
When dividing terms with the same base, subtract the denominator's exponent from the numerator's exponent.
3
Simplify the third expression by applying the power of a product rule.
2^4 = 16, a^(3*4) = a^12, b^(-2*4) = b^-8. Combined: 16a^12b^-8
When raising a power to a power, multiply the exponents. Apply the exponent to the numerical coefficient as well.
4
Simplify the fourth expression by applying the zero exponent rule, then the power rule.
a^0 = 1. 5^-2 = 1/25. b^(3*-2) = b^-6. Combined: 1/(25b^6)
Any non-zero term raised to the power of 0 is 1. Negative exponents indicate taking the reciprocal.

Key Concept

Applying the product, quotient, power, zero, and negative rules of exponents to simplify algebraic expressions with multiple variables and coefficients.
Question 30Question →

Consider the algebraic expression below, where xx and yy represent non-zero values:

(4x3y−2)22x−1y3 \frac{(4x^3 y^{-2})^2}{2x^{-1} y^3}

Which of the following represents the completely simplified form of this expression?

Show answer & explanation

Answer: 8x7y7 \frac{8x^7}{y^7}

Answer

The completely simplified form is 8x7y7 \frac{8x^7}{y^7} .
The expression is simplified by first distributing the outer exponent of 2 to every factor in the numerator, which squares the coefficient 4 and multiplies the variables' exponents to produce 16x6y−416x^6y^{-4}. Next, dividing the coefficients yields 16÷2=816 \div 2 = 8. Then, the quotient rule is applied by subtracting the denominator's exponents from the numerator's exponents for matching bases: for xx, 6−(−1)=76 - (-1) = 7, and for yy, −4−3=−7-4 - 3 = -7. This results in 8x7y−78x^7y^{-7}. Finally, the negative exponent is resolved by moving y7y^7 to the denominator, giving the completely simplified form 8x7y7 \frac{8x^7}{y^7} .

Step-by-Step Solution

1
Apply the power of a product rule to simplify the numerator.
(4x3y−2)2=42⋅(x3)2⋅(y−2)2=16x6y−4 (4x^3 y^{-2})^2 = 4^2 \cdot (x^3)^2 \cdot (y^{-2})^2 = 16x^6y^{-4}
When an algebraic term is raised to a power, the outer exponent multiplies the exponent of each factor inside the parentheses, including the numerical coefficient (which has an implied exponent of 1).
2
Rewrite the fraction using the expanded numerator.
16x6y−42x−1y3 \frac{16x^6y^{-4}}{2x^{-1} y^3}
This sets up the expression to cleanly apply the quotient rules for exponents and to divide the numerical coefficients.
3
Divide the numerical coefficients.
16÷2=8 16 \div 2 = 8
Coefficients follow standard arithmetic division rules independent of the variables.
4
Apply the quotient rule to the variables by subtracting the denominator's exponents from the numerator's exponents.
For xx: 6−(−1)=6+1=7 6 - (-1) = 6 + 1 = 7 . For yy: −4−3=−7 -4 - 3 = -7 .
The quotient rule states that aman=am−n \frac{a^m}{a^n} = a^{m-n} . Note that subtracting a negative number is algebraically equivalent to addition.
5
Combine the terms and convert the negative exponent into a positive exponent.
8x7y−7=8x7y7 8x^7y^{-7} = \frac{8x^7}{y^7}
A negative exponent represents the reciprocal of the base (a−n=1an a^{-n} = \frac{1}{a^n} ). Moving y−7y^{-7} to the denominator as y7y^7 yields the final simplified expression format.

Key Concept

Operations with Variable Exponents
Question 31Question →
When the algebraic expression
(3x−2y4)39x−5y2 \frac{(3x^{-2} y^4)^3}{9x^{-5} y^2}
is fully simplified to the form CxaybC x^a y^b (where CC, aa, and bb are constants), what is the value of bb?
Show answer & explanation

Answer: 10; ten

Answer

10
By distributing the outer exponent of 3 in the numerator using multiplication (4⋅34 \cdot 3) to get y12y^{12}, and then subtracting the denominator's exponent of 2 using the quotient rule (12−212 - 2), the final exponent on the variable yy is determined to be 10.

Step-by-Step Solution

1
Apply the power of a product and power of a power rules to the numerator.
(3x−2y4)3=33⋅(x−2)3⋅(y4)3=27x−6y12(3x^{-2}y^4)^3 = 3^3 \cdot (x^{-2})^3 \cdot (y^4)^3 = 27x^{-6}y^{12}
When raising a product to a power, the outer power applies to each individual factor inside the parentheses. Exponents are multiplied when raising a power to another power.
2
Substitute the expanded numerator back into the fractional expression.
27x−6y129x−5y2 \frac{27x^{-6}y^{12}}{9x^{-5}y^2}
This sets up the expression for applying the quotient rule between the numerator and denominator.
3
Apply the quotient rule by dividing the coefficients and subtracting the exponents of like bases.
Coefficient: 27÷9=327 \div 9 = 3
Base xx: x−6−(−5)=x−6+5=x−1x^{-6 - (-5)} = x^{-6 + 5} = x^{-1}
Base yy: y12−2=y10y^{12 - 2} = y^{10}
When dividing exponential terms with the same base, subtract the denominator's exponent from the numerator's exponent.
4
Combine the simplified terms and identify the exponent bb.
The simplified expression is 3x−1y103x^{-1}y^{10}. Comparing this to the target form CxaybCx^a y^b, the value of bb is 1010.
This isolates the specific value requested by the question stem.

Key Concept

Power of a Power and Quotient of Powers Rules
Question 32Question →

An algebraic expression is given as:

(3a4b−2)2(2a−3b5)6a2b \frac{(3a^4b^{-2})^2 (2a^{-3}b^5)}{6a^2b}

What is the fully simplified form of this expression for all non-zero values of aa and bb?

Show answer & explanation

Answer: 3a33a^3

Answer

The simplified form of the expression is 3a33a^3.
The expression is correctly simplified by first applying the power to all terms inside the parentheses (resulting in 9a8b−49a^8b^{-4}), then multiplying by the adjacent term in the numerator (resulting in 18a5b18a^5b), and finally dividing by the denominator (giving 3a33a^3).

Step-by-Step Solution

1
Simplify the squared term in the numerator using the power of a product and power of a power rules.
9a8b−49a^8b^{-4}
When a product is raised to a power, raise each factor to that power (32=93^2 = 9). For variables, multiply the inner and outer exponents: (a4)2=a8(a^4)^2 = a^8 and (b−2)2=b−4(b^{-2})^2 = b^{-4}.
2
Multiply the simplified term by the second factor in the numerator, (2a−3b5)(2a^{-3}b^5).
18a5b18a^5b
Multiply the coefficients (9×2=189 \times 2 = 18). For variables with the same base, add their exponents: a8+(−3)=a5a^{8+(-3)} = a^5 and b−4+5=b1=bb^{-4+5} = b^1 = b.
3
Divide the resulting numerator by the denominator, 6a2b6a^2b.
3a33a^3
Divide the coefficients (18÷6=318 \div 6 = 3). For variables with the same base, subtract the denominator's exponent from the numerator's exponent: a5−2=a3a^{5-2} = a^3. The terms b1b^1 and b1b^1 cancel out since b1−1=b0=1b^{1-1} = b^0 = 1.

Key Concept

Operations with Variable Exponents
Question 33Question →

The length of a rectangular solar panel is given by the expression 7a4b−27a^4 b^{-2} and its width is given by 6a2b56a^2 b^5. Which of the following expressions represents the area of the solar panel?

Show answer & explanation

Answer: 42a6b342a^6 b^3

Answer

The area is represented by the expression 42a6b342a^6 b^3.
The correct answer is 42a6b342a^6 b^3. To find the area of the rectangle, you multiply its length and width: (7a4b−2)(6a2b5)(7a^4 b^{-2})(6a^2 b^5). The coefficients are multiplied together (7×6=427 \times 6 = 42). For the variables, the product rule states that when multiplying terms with the same base, you add their exponents. Thus, for base aa, 4+2=64 + 2 = 6 to get a6a^6, and for base bb, −2+5=3-2 + 5 = 3 to get b3b^3. Combining these yields 42a6b342a^6 b^3.

Step-by-Step Solution

1
Identify the formula for the area of a rectangle and set up the expression.
Area = length \times width = (7a^4 b^{-2})(6a^2 b^5)
The area of a rectangle is found by taking the product of its length and width dimensions.
2
Multiply the numerical coefficients.
7×6=427 \times 6 = 42
Constants must be multiplied independently from the variable bases.
3
Apply the product rule for exponents to the variables with identical bases.
a^{4+2} = a^6 and b^{-2+5} = b^3
When multiplying identical bases, their exponents must be added.
4
Combine the calculated coefficient and variables into a single expression.
42a6b342a^6 b^3
This forms the final simplified product representing the area.

Key Concept

Operations with Variable Exponents
Estimated Time:1m 0s
Question 34Question →

An engineer is calculating a scaling factor for a fluid dynamics simulation. The calculation reduces to the following algebraic expression:

(3m3n−4)(8m−1n2)(2m−2n3)2 \frac{ (3m^3n^{-4})(8m^{-1}n^2) }{ (2m^{-2}n^3)^2 }

Assuming mm and nn are non-zero variables, what is the fully simplified form of this scaling factor?

Show answer & explanation

Answer: 6m6n8\frac{6m^6}{n^8}

Answer

The fully simplified expression is 6m6n8\frac{6m^6}{n^8}.
The correct answer is found by simplifying the numerator and denominator independently, then dividing. Multiplying the terms in the numerator gives 24m2n−224m^2n^{-2}. Squaring the entire term in the denominator (including the 2) gives 4m−4n64m^{-4}n^6. Dividing these expressions yields a coefficient of 66. Subtracting the exponents gives m2−(−4)=m6m^{2 - (-4)} = m^6 and n−2−6=n−8n^{-2 - 6} = n^{-8}. Finally, converting the negative exponent into a fraction results in 6m6n8\frac{6m^6}{n^8}.

Step-by-Step Solution

1
Simplify the numerator using the product rule for exponents.
(3×8)m3+(−1)n−4+2=24m2n−2(3 \times 8) m^{3 + (-1)} n^{-4 + 2} = 24m^2n^{-2}
When multiplying terms with the same base, add their exponents and multiply their numerical coefficients.
2
Simplify the denominator using the power of a product rule.
22(m−2×2)(n3×2)=4m−4n62^2(m^{-2 \times 2})(n^{3 \times 2}) = 4m^{-4}n^6
When raising a product to a power, raise the coefficient and multiply every exponent by the outside power.
3
Divide the simplified numerator by the simplified denominator using the quotient rule.
(244)m2−(−4)n−2−6=6m6n−8(\frac{24}{4}) m^{2 - (-4)} n^{-2 - 6} = 6m^6n^{-8}
When dividing terms with the same base, subtract the denominator's exponent from the numerator's exponent.
4
Rewrite the expression with only positive exponents.
6m6n8\frac{6m^6}{n^8}
A negative exponent means taking the reciprocal of the base, so n−8n^{-8} becomes 1n8\frac{1}{n^8}.

Key Concept

Operations with Variable Exponents
Question 35Question →

Match each algebraic expression on the left with its fully simplified equivalent form on the right. Assume all variables represent non-zero real numbers.

Click a left item, then click its matching right item

Items

15a4b−35a−2b2\frac{15a^4b^{-3}}{5a^{-2}b^2}
(3a−2b4)2(3a^{-2}b^4)^2
(2a3b−1)(4a−1b3)(2a^3b^{-1})(4a^{-1}b^3)
(a2b−22)−3\left(\frac{a^2b^{-2}}{2}\right)^{-3}

Matches

Show answer & explanation

Answer

The quotient matches (3a^6)/b^5, the squared term matches (9b^8)/a^4, the product matches 8a^2b^2, and the fraction raised to a negative power matches (8b^6)/a^6.
Each expression is thoroughly simplified by applying the fundamental laws of exponents: adding exponents for multiplication, subtracting exponents for division, multiplying exponents for a power to a power, and reciprocating terms to handle negative exponents.

Step-by-Step Solution

1
Simplify the first expression: (15a^4b^-3) / (5a^-2b^2).
3a^6b^-5, which becomes (3a^6)/b^5.
When dividing terms with the same base, divide the coefficients and subtract the exponents. Move negative exponents to the denominator.
2
Simplify the second expression: (3a^-2b^4)^2.
9a^-4b^8, which becomes (9b^8)/a^4.
When raising a power to a power, raise the coefficient to the outer power and multiply the variable exponents.
3
Simplify the third expression: (2a^3b^-1)(4a^-1b^3).
8a^2b^2.
When multiplying terms with the same base, multiply the coefficients and add the respective exponents.
4
Simplify the fourth expression: ((a^2b^-2)/2)^-3.
a^-6b^6 / 2^-3, which becomes (8b^6)/a^6.
Apply the outer negative exponent to all factors inside the parentheses, including the denominator, then move negative powers across the fraction bar to make them positive.

Key Concept

Exponent rules: Product of powers, quotient of powers, power of a product, and negative exponents.
Question 36Question →
The algebraic expression (2x4y3)34x5y−2\frac{(2x^4 y^3)^3}{4x^5 y^{-2}} can be simplified to the form kxaybkx^a y^b, where kk is the numerical coefficient, and aa and bb are the exponents. What is the value of a+ba + b?
Show answer & explanation

Answer: 18; 18.0

Answer

18
By applying the power rule, the numerator becomes 8x12y98x^{12}y^9. Dividing this by the denominator 4x5y−24x^5y^{-2} yields a numerical coefficient of 8/4=28/4=2. For the xx terms, applying the quotient rule gives 12−5=712 - 5 = 7. For the yy terms, 9−(−2)=119 - (-2) = 11. The simplified expression is 2x7y112x^7y^{11}. The exponents are a=7a=7 and b=11b=11, so their sum is 7+11=187 + 11 = 18.

Step-by-Step Solution

1
Apply the power of a product rule to the numerator.
(23)(x4×3)(y3×3)=8x12y9(2^3)(x^{4 \times 3})(y^{3 \times 3}) = 8x^{12}y^9
To distribute the outer exponent 3 to all factors inside the parentheses.
2
Divide the numerical coefficients.
84=2\frac{8}{4} = 2
To simplify the constants to find kk.
3
Apply the quotient rule for the variable xx.
x12−5=x7x^{12 - 5} = x^7
When dividing like bases, subtract the exponent in the denominator from the exponent in the numerator. This gives a=7a = 7.
4
Apply the quotient rule for the variable yy, accounting for the negative exponent.
y9−(−2)=y9+2=y11y^{9 - (-2)} = y^{9 + 2} = y^{11}
Subtracting a negative exponent is equivalent to adding. This gives b=11b = 11.
5
Calculate the sum of the exponents aa and bb.
7+11=187 + 11 = 18
To answer the specific question asked in the prompt.

Key Concept

Applying the power of a product rule and the quotient rule for exponents, paying special attention to subtracting negative exponents.
Question 37Question →
Assume xx and yy are non-zero real numbers. If the algebraic expression (3x2y4)39xky7 \frac{(3x^2y^4)^3}{9x^ky^7} simplifies completely to 3xy5 3xy^5 , what is the value of the exponent kk?
Show answer & explanation

Answer: 5; k=5; k = 5

Answer

5
The correct value is 5. By expanding the numerator using the power of a product rule, (3x2y4)3(3x^2y^4)^3 becomes 27x6y1227x^6y^{12}. Dividing this by the denominator 9xky79x^ky^7 yields (279)⋅x6−k⋅y12−7\left(\frac{27}{9}\right) \cdot x^{6-k} \cdot y^{12-7}, which simplifies to 3x6−ky53x^{6-k}y^5. Equating this result to the given simplified expression 3x1y53x^1y^5 means the exponent of xx must be 1. Solving the simple equation 6−k=16 - k = 1 gives k=5k = 5.

Step-by-Step Solution

1
Expand the numerator using the power of a product and power of a power rules.
(3x2y4)3=33⋅(x2)3⋅(y4)3=27x6y12(3x^2y^4)^3 = 3^3 \cdot (x^2)^3 \cdot (y^4)^3 = 27x^6y^{12}
The outer exponent must be applied to the numerical coefficient and multiplied by the inner exponents of each variable.
2
Substitute the expanded numerator back into the fractional expression.
27x6y129xky7 \frac{27x^6y^{12}}{9x^ky^7}
This sets up the expression for division.
3
Simplify the numerical coefficients and apply the quotient rule for exponents (xa/xb=xa−bx^a / x^b = x^{a-b}).
(279)⋅x6−k⋅y12−7=3x6−ky5\left(\frac{27}{9}\right) \cdot x^{6-k} \cdot y^{12-7} = 3x^{6-k}y^5
To combine the numerical terms and the variable terms with identical bases.
4
Set the simplified expression equal to the target expression provided in the problem.
3x6−ky5=3x1y53x^{6-k}y^5 = 3x^1y^5
The problem states that the original expression simplifies completely to 3xy53xy^5.
5
Equate the exponents of the base xx to solve for kk.
6−k=1⇒k=56 - k = 1 \Rightarrow k = 5
Since the expressions are equal and the bases are the same, their corresponding exponents must be equal.

Key Concept

Applying power and quotient rules of exponents to solve for an unknown variable exponent.
Question 38Question →

Simplify the following algebraic expression:

(12p4q−33p−2q2)−1 \left( \frac{12p^4 q^{-3}}{3p^{-2} q^2} \right)^{-1}

What is the fully simplified form of this expression?

Show answer & explanation

Answer: q54p6\frac{q^5}{4p^6}

Answer

The expression simplifies to q54p6\frac{q^5}{4p^6}.
The correct answer is derived by first simplifying the inner fraction using the quotient rule for exponents, resulting in 4p6q−54p^6q^{-5}. Applying the outer negative exponent of −1-1 takes the reciprocal of each factor, leading to 4−1p−6q54^{-1}p^{-6}q^5, which is written with positive exponents as q54p6\frac{q^5}{4p^6}.

Step-by-Step Solution

1
Divide the numerical coefficients.
12÷3=412 \div 3 = 4
Coefficients follow standard division rules.
2
Simplify the variable pp using the quotient rule for exponents.
p4÷p−2=p4−(−2)=p6p^4 \div p^{-2} = p^{4 - (-2)} = p^6
When dividing like bases, subtract the exponent in the denominator from the exponent in the numerator.
3
Simplify the variable qq using the quotient rule for exponents.
q−3÷q2=q−3−2=q−5q^{-3} \div q^2 = q^{-3 - 2} = q^{-5}
Subtracting a positive exponent is mathematically equivalent to adding its negative value.
4
Combine the simplified parts into a single expression inside the parentheses.
4p6q−54p^6q^{-5}
This represents the fully simplified value of the fraction before applying the outer exponent.
5
Apply the outer exponent of −1-1 to every factor in the expression.
(4p6q−5)−1=4−1⋅(p6)−1⋅(q−5)−1(4p^6q^{-5})^{-1} = 4^{-1} \cdot (p^6)^{-1} \cdot (q^{-5})^{-1}
The power of a product rule states that (abc)x=axbxcx(abc)^x = a^x b^x c^x.
6
Simplify the resulting exponents and write with positive exponents.
4−1p−6q5=q54p64^{-1}p^{-6}q^5 = \frac{q^5}{4p^6}
A negative exponent indicates the reciprocal of the base (x−n=1/xnx^{-n} = 1/x^n).

Key Concept

Simplifying algebraic fractions using the quotient, product, and negative power rules for exponents.
Question 39Question →

Determine the simplified form of the following product, assuming cc and dd are non-zero real numbers:

(2c−3d5)−2(4c2d−1)(2c^{-3}d^5)^{-2}(4c^2d^{-1})

What is the resulting fully simplified expression?

Show answer & explanation

Answer: c8d11\frac{c^8}{d^{11}}

Answer

The fully simplified expression is c8d11\frac{c^8}{d^{11}}.
The correct answer is obtained by first applying the power of a power rule to (2c−3d5)−2(2c^{-3}d^5)^{-2}, yielding 14c6d−10\frac{1}{4}c^6d^{-10}. Multiplying this by 4c2d−14c^2d^{-1} involves adding the exponents of like bases, which results in c8d−11c^8d^{-11}. Moving the negative exponent to the denominator yields the fully simplified form.

Step-by-Step Solution

1
Apply the power of a product rule to the first term, distributing the −2-2 exponent to the coefficient and both variables.
2−2(c−3)−2(d5)−2=14c6d−102^{-2}(c^{-3})^{-2}(d^5)^{-2} = \frac{1}{4}c^6d^{-10}
The rules (xy)a=xaya(xy)^a = x^ay^a and (xa)b=xab(x^a)^b = x^{ab} require every part of the base to be raised to the outer power.
2
Multiply the expanded first term by the second term by grouping the coefficients and the like variables together.
(14⋅4)(c6⋅c2)(d−10⋅d−1)(\frac{1}{4} \cdot 4)(c^6 \cdot c^2)(d^{-10} \cdot d^{-1})
Multiplication is commutative, which allows rearranging the terms to combine matching bases.
3
Multiply the numerical coefficients and add the exponents for the variables with the same base.
1⋅c6+2⋅d−10+(−1)=c8d−111 \cdot c^{6+2} \cdot d^{-10+(-1)} = c^8d^{-11}
The product of powers rule states that when multiplying terms with the same base, their exponents should be added (xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}).
4
Rewrite the expression so that all exponents are positive.
c8d11\frac{c^8}{d^{11}}
A fully simplified algebraic expression is typically written with positive exponents, using the rule x−a=1xax^{-a} = \frac{1}{x^a}.

Key Concept

Operations with Variable Exponents
Question 40Question →

Match each algebraic expression containing numerical coefficients on the left with its fully simplified equivalent form on the right.

Click a left item, then click its matching right item

Items

3y2⋅2y43y^2 \cdot 2y^4
12y74y3\frac{12y^7}{4y^3}
(2y3)3(2y^3)^3
5y−25y^{-2}

Matches

Show answer & explanation

Answer

The simplified pairs are: 3y2⋅2y43y^2 \cdot 2y^4 matches with 6y66y^6, 12y74y3\frac{12y^7}{4y^3} matches with 3y43y^4, (2y3)3(2y^3)^3 matches with 8y98y^9, and 5y−25y^{-2} matches with 5y2\frac{5}{y^2}.
Each left-side expression simplifies to exactly one unique form on the right by appropriately applying exponent rules to the variable bases and standard arithmetic operations to the numerical coefficients.

Step-by-Step Solution

1
Simplify the product expression 3y2⋅2y43y^2 \cdot 2y^4.
6y66y^6
When multiplying terms, handle coefficients and variables separately. Multiply coefficients (3×2=63 \times 2 = 6) and add exponents for identical bases (y2+4=y6y^{2+4} = y^6).
2
Simplify the quotient expression 12y74y3\frac{12y^7}{4y^3}.
3y43y^4
Divide the coefficients (12÷4=312 \div 4 = 3) and subtract the bottom exponent from the top exponent (y7−3=y4y^{7-3} = y^4).
3
Simplify the power expression (2y3)3(2y^3)^3.
8y98y^9
Raise both the coefficient and the variable to the power of 3. Calculate 23=82^3 = 8, and multiply the exponents for the variable 3×3=93 \times 3 = 9.
4
Simplify the negative exponent expression 5y−25y^{-2}.
5y2\frac{5}{y^2}
A negative exponent creates a reciprocal. Since the exponent only applies to yy, it moves to the denominator while the coefficient 5 stays in the numerator.

Key Concept

Handling numerical coefficients alongside basic operations with variable exponents.
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Operations with Variable Exponents Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 2 | Examkin