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Operations with Variable Exponents

43 questions

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Question 41Question →

Simplify the following algebraic expression completely:

(5x−2y4)(−2x5y−1)2 (5x^{-2}y^4)(-2x^5y^{-1})^2

Which of the following represents the correct equivalent form?

Show answer & explanation

Answer: 20x8y220x^8y^2

Answer

The correctly simplified expression is 20x8y220x^8y^2.
The correct result is achieved by following the order of operations and applying the laws of exponents. First, the exponent of 2 must be distributed to all parts of the second term: (−2)2(x5)2(y−1)2(-2)^2(x^5)^2(y^{-1})^2, which simplifies to 4x10y−24x^{10}y^{-2}. Next, this result is multiplied by the first term: (5x−2y4)(4x10y−2)(5x^{-2}y^4)(4x^{10}y^{-2}). Multiplying the coefficients gives 5×4=205 \times 4 = 20. Using the product of powers rule, adding the exponents for xx yields −2+10=8-2 + 10 = 8, and for yy yields 4+(−2)=24 + (-2) = 2. This results in the final expression 20x8y220x^8y^2.

Step-by-Step Solution

1
Apply the power of a product rule to the second term by squaring the coefficient and each variable.
(−2)2(x5)2(y−1)2=4x10y−2(-2)^2(x^5)^2(y^{-1})^2 = 4x^{10}y^{-2}
The order of operations (PEMDAS) requires evaluating exponents before performing multiplication.
2
Rewrite the full expression with the expanded term.
(5x−2y4)(4x10y−2)(5x^{-2}y^4)(4x^{10}y^{-2})
To clearly identify the coefficients and bases that need to be combined.
3
Multiply the numerical coefficients and apply the product of powers rule to the variables by adding their exponents.
(5×4)x−2+10y4+(−2)=20x8y2(5 \times 4)x^{-2+10}y^{4+(-2)} = 20x^8y^2
When multiplying exponential terms with the same base, you add the exponents.

Key Concept

Operations with Variable Exponents
Question 42Question →

Pair each algebraic operation on the left with its correctly simplified expression on the right. Assume all variables represent non-zero real numbers.

Click a left item, then click its matching right item

Items

(3m4n2)(4m2n5)(3m^4 n^2)(4m^2 n^5)
12m8n34m2n−4\frac{12m^8 n^3}{4m^2 n^{-4}}
(2m3n2)3(2m^3 n^2)^3
24m4n23m5n−1\frac{24m^4 n^2}{3m^5 n^{-1}}

Matches

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Answer

The correct matches require applying the product, quotient, and power rules of exponents systematically to each expression.
Each expression is correctly simplified by identifying the primary exponent rule required (product, quotient, or power) and applying it to both the numerical coefficients and the variables' exponents.

Step-by-Step Solution

1
Evaluate (3m4n2)(4m2n5)(3m^4 n^2)(4m^2 n^5) using the Product Rule.
12m6n712m^6 n^7
Multiply coefficients (3 * 4 = 12) and add exponents of corresponding variables (m4+2m^{4+2}, n2+5n^{2+5}).
2
Evaluate 12m8n34m2n−4\frac{12m^8 n^3}{4m^2 n^{-4}} using the Quotient Rule.
3m6n73m^6 n^7
Divide coefficients (12 / 4 = 3) and subtract denominator exponents from numerator exponents (m8−2m^{8-2}, n3−(−4)n^{3-(-4)}).
3
Evaluate (2m3n2)3(2m^3 n^2)^3 using the Power of a Product Rule.
8m9n68m^9 n^6
Cube the coefficient (23=82^3 = 8) and multiply inner exponents by the outer exponent (m3×3m^{3 \times 3}, n2×3n^{2 \times 3}).
4
Evaluate 24m4n23m5n−1\frac{24m^4 n^2}{3m^5 n^{-1}} using the Quotient Rule and Negative Exponent Rule.
8n3m\frac{8n^3}{m}
Divide coefficients (24 / 3 = 8), subtract exponents (m4−5=m−1m^{4-5} = m^{-1}, n2−(−1)=n3n^{2-(-1)} = n^3), and rewrite the negative exponent in the denominator.

Key Concept

Applying fundamental exponent laws (product rule, quotient rule, power of a power) and properly handling numerical coefficients separately from variable exponents.
Estimated Time:1m 30s
Question 43Question →

Which of the following is equivalent to the expression (3m−2n4)2(2m5n0)(3m^{-2}n^4)^2(2m^5n^0)?

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Answer: 18mn818mn^8

Answer

The simplified expression is 18mn818mn^8.
The correct answer is derived by first squaring each part of the initial term, making it 9m−4n89m^{-4}n^8. The second term simplifies to 2m52m^5 because n0=1n^0 = 1. Multiplying these together requires multiplying the coefficients (9×2=189 \times 2 = 18) and adding the exponents for mm (−4+5=1-4 + 5 = 1), resulting in 18mn818mn^8.

Step-by-Step Solution

1
Simplify the first factor by applying the power of 2 to the coefficient and the variable exponents.
(3)2(m−2)2(n4)2=9m−4n8(3)^2(m^{-2})^2(n^4)^2 = 9m^{-4}n^8
When a product is raised to a power, the exponent must be applied to every factor inside the parentheses, multiplying the variable exponents.
2
Simplify the second factor using the zero exponent rule.
2m5(1)=2m52m^5(1) = 2m^5
Any non-zero term raised to the power of 0 equals 1, so n0=1n^0 = 1.
3
Multiply the two simplified factors together.
(9m−4n8)(2m5)=(9⋅2)(m−4⋅m5)(n8)(9m^{-4}n^8)(2m^5) = (9 \cdot 2)(m^{-4} \cdot m^5)(n^8)
To multiply terms, multiply the numerical coefficients and add the exponents of like bases.
4
Perform the final arithmetic for the coefficients and exponents.
18m−4+5n8=18m1n8=18mn818m^{-4+5}n^8 = 18m^1n^8 = 18mn^8
Evaluating 9⋅2=189 \cdot 2 = 18 and combining the exponents of mm gives the final simplified expression.

Key Concept

Operations with Variable Exponents
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