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Tüm alıştırma soruları

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Soru 4921Soru →

Which of the following is equivalent to the algebraic expression below after it has been fully simplified?

4k(k−3m)−(2k2−5km+m2)4k(k - 3m) - (2k^2 - 5km + m^2)
Cevabı ve açıklamayı göster

Cevap: 2k2−7km−m22k^2 - 7km - m^2

Cevap

The fully simplified expression is 2k2−7km−m22k^2 - 7km - m^2.
The correct expression is derived by first multiplying 4k4k by both terms in (k−3m)(k - 3m) to get 4k2−12km4k^2 - 12km. Next, distributing the negative sign through the trinomial changes the signs to yield −2k2+5km−m2-2k^2 + 5km - m^2. Finally, grouping and combining like terms yields 4k2−2k2=2k24k^2 - 2k^2 = 2k^2 and −12km+5km=−7km-12km + 5km = -7km. The −m2-m^2 term has no like terms to combine with, leaving it unchanged. The final simplified form is 2k2−7km−m22k^2 - 7km - m^2.

Adım Adım Çözüm

1
Distribute 4k4k into the binomial (k−3m)(k - 3m).
4k2−12km4k^2 - 12km
Expanding the first part of the expression removes the first set of parentheses.
2
Distribute the negative sign across the trinomial (2k2−5km+m2)(2k^2 - 5km + m^2).
−2k2+5km−m2-2k^2 + 5km - m^2
Subtracting a polynomial requires reversing the sign of every term inside the parentheses.
3
Group and combine all like terms.
(4k2−2k2)+(−12km+5km)−m2=2k2−7km−m2(4k^2 - 2k^2) + (-12km + 5km) - m^2 = 2k^2 - 7km - m^2
Expressions are fully simplified when all terms with identical variable combinations are merged.

Anahtar Kavram

Distributing variables and negative signs followed by combining like terms.
Soru 4922Soru →

If the equation y8⋅yk=y15y^8 \cdot y^k = y^{15} is true for all non-zero values of yy, what is the value of kk?

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Cevap: 7; seven

Cevap

7
The correct value of kk is 77 because applying the product rule of exponents gives y8⋅y7=y8+7=y15y^8 \cdot y^7 = y^{8+7} = y^{15}.

Adım Adım Çözüm

1
Apply the product rule for exponents to the left side of the equation.
y8⋅yk=y8+ky^8 \cdot y^k = y^{8+k}
When multiplying exponential expressions with the same base, the exponents are added.
2
Set the exponent of the simplified left side equal to the exponent of the right side.
8+k=158 + k = 15
Since the bases are equal and non-zero, their exponents must be equal.
3
Solve the linear equation for kk.
k=7k = 7
Subtracting 8 from both sides of the equation isolates the variable kk.

Anahtar Kavram

Product Rule for Exponents
Soru 4923Soru →

A student is required to identify a calculation that yields a value greater than zero. Which of the following arithmetic expressions produces a positive integer result?

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Cevap: (−3)×(−4)−10(-3) \times (-4) - 10

Cevap

The expression (−3)×(−4)−10(-3) \times (-4) - 10 produces a positive integer.
The expression (−3)×(−4)−10(-3) \times (-4) - 10 evaluates to a positive integer because the product of −3-3 and −4-4 is a positive 1212, and subtracting 1010 from 1212 leaves 22.

Adım Adım Çözüm

1
Evaluate the first expression: −5+(−2)-5 + (-2).
−5+(−2)=−7-5 + (-2) = -7.
Adding a negative integer to a negative integer results in a smaller negative integer.
2
Evaluate the second expression: 4−3×24 - 3 \times 2.
4−6=−24 - 6 = -2.
Following the order of operations (PEMDAS), multiplication is performed before subtraction.
3
Evaluate the third expression: −∣−8∣+5-|-8| + 5.
−8+5=−3-8 + 5 = -3.
The absolute value of −8-8 is positive 88. The negative sign outside makes it −8-8. Adding 55 results in a negative integer.
4
Evaluate the fourth expression: (−3)×(−4)−10(-3) \times (-4) - 10.
12−10=212 - 10 = 2.
Multiplying two negative integers yields a positive integer (1212). Subtracting 1010 from 1212 leaves a positive integer.

Anahtar Kavram

Integer operations combining sign rules, absolute value, and the order of operations.
Tahmini Süre:50s
Soru 4924Soru →

Simplify the following algebraic expression completely:

(3a5)2(3a^5)^2

What is the simplified form of this expression?

Cevabı ve açıklamayı göster

Cevap: 9a109a^{10}

Cevap

The simplified expression is 9a109a^{10}.
The correct result is found by distributing the exponent of 2 to both factors inside the parentheses. The coefficient 3 is squared to become 9, and the exponent 5 is multiplied by 2 to become 10, resulting in 9a109a^{10}.

Adım Adım Çözüm

1
Apply the power of a product rule by distributing the outer exponent to both the numerical coefficient and the variable inside the parentheses.
32⋅(a5)23^2 \cdot (a^5)^2
When a product is raised to a power, each independent factor inside the parentheses must be raised to that power.
2
Simplify the numerical coefficient by calculating its square.
9⋅(a5)29 \cdot (a^5)^2
The exponent 2 indicates multiplying the base by itself: 3×3=93 \times 3 = 9.
3
Apply the power of a power rule to the variable by multiplying the inner exponent by the outer exponent.
9a109a^{10}
According to exponent laws, (xm)n=xm⋅n(x^m)^n = x^{m \cdot n}, so the new exponent is 5×2=105 \times 2 = 10.

Anahtar Kavram

Power of a Product and Power of a Power Rules
Soru 4925Soru →

What is the value of the following expression?

8.4−2.4×1.5+4.5÷0.58.4 - 2.4 \times 1.5 + 4.5 \div 0.5
Cevabı ve açıklamayı göster

Cevap: 13.8

Cevap

13.8
Applying the order of operations correctly requires first evaluating the multiplication (2.4×1.5=3.62.4 \times 1.5 = 3.6) and the division (4.5÷0.5=94.5 \div 0.5 = 9). Substituting these values back simplifies the equation to 8.4−3.6+98.4 - 3.6 + 9. Processing the remaining subtraction and addition from left to right yields 4.8+9=13.84.8 + 9 = 13.8.

Adım Adım Çözüm

1
Evaluate the multiplication part of the expression.
2.4×1.5=3.62.4 \times 1.5 = 3.6
According to the order of operations, multiplication must be performed before addition or subtraction.
2
Evaluate the division part of the expression.
4.5÷0.5=94.5 \div 0.5 = 9
Division shares the same precedence as multiplication and must also be evaluated before addition and subtraction.
3
Substitute the results back into the original expression.
8.4−3.6+98.4 - 3.6 + 9
This simplifies the expression so that only addition and subtraction remain.
4
Evaluate the remaining addition and subtraction from left to right.
8.4−3.6=4.88.4 - 3.6 = 4.8, and 4.8+9=13.84.8 + 9 = 13.8
Addition and subtraction have equal precedence and are solved strictly from left to right.

Anahtar Kavram

Order of Operations with Decimals
Soru 4926Soru →

An inventory manager calculates the total number of pallets, hh, that can be stored in a warehouse section using the following linear equation:

3h+6=453h + 6 = 45

What is the value of hh?

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Cevap: 13

Cevap

The correct value for the variable is 13.
The correct value is found by performing the inverse operations in the correct order. First, subtract 6 from both sides to get 3h = 39. Then, divide both sides by 3 to find that h = 13.

Adım Adım Çözüm

1
Isolate the term with the variable by subtracting 6 from both sides of the equation.
3h=45−63h = 45 - 6
3h=393h = 39
This removes the constant term from the left side, keeping the equation balanced.
2
Solve for hh by dividing both sides of the equation by 3.
h=393h = \frac{39}{3}
h=13h = 13
Dividing by the coefficient of the variable isolates hh completely.

Anahtar Kavram

Solving basic two-step linear equations by using inverse operations to isolate the variable.
Soru 4927Soru →

A specialized aerospace alloy is manufactured by combining titanium and aluminum in a ratio of 7:47:4 by weight. If a batch of this alloy contains 140 kilograms140\text{ kilograms} of titanium, what is the total weight of the batch?

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Cevap: 220 kg220\text{ kg}

Cevap

The total weight of the batch is 220 kg220\text{ kg}.
The given ratio indicates the alloy consists of 77 parts titanium and 44 parts aluminum, making 1111 parts in total. Because the 77 parts of titanium weigh 140 kg140\text{ kg}, each single part weighs 20 kg20\text{ kg} (140÷7140 \div 7). Multiplying the 1111 total parts by 20 kg20\text{ kg} per part results in a total weight of 220 kg220\text{ kg}.

Adım Adım Çözüm

1
Determine the weight of a single ratio part based on the known titanium content.
140 kg÷7 parts=20 kg per part140\text{ kg} \div 7\text{ parts} = 20\text{ kg per part}
Since the ratio of titanium to aluminum is 7:47:4, the 140 kg140\text{ kg} corresponds exactly to the 77 parts of titanium.
2
Calculate the total number of parts in the alloy mixture.
7 parts (titanium)+4 parts (aluminum)=11 total parts7\text{ parts (titanium)} + 4\text{ parts (aluminum)} = 11\text{ total parts}
The total weight is composed of both the titanium and the aluminum parts combined.
3
Multiply the total number of parts by the weight of a single part.
11 parts×20 kg per part=220 kg11\text{ parts} \times 20\text{ kg per part} = 220\text{ kg}
This yields the combined weight of the entire batch.

Anahtar Kavram

Solving Part-to-Whole Ratio Problems
Tahmini Süre:1m 0s
Soru 4928Soru →

Solve the following equation for pp:

23(6p−15)+3p=−3(p−8)+2p−10 \frac{2}{3}(6p - 15) + 3p = -3(p - 8) + 2p - 10

What is the final value of pp?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The correct value for the variable is 3.
The correct answer is 3. Distributing the terms on both sides yields 4p−10+3p=−3p+24+2p−104p - 10 + 3p = -3p + 24 + 2p - 10. Combining like terms results in 7p−10=−p+147p - 10 = -p + 14. Adding pp to both sides and adding 10 to both sides gives 8p=248p = 24. Dividing by 8 isolates the variable, resulting in p=3p = 3.

Adım Adım Çözüm

1
Distribute the coefficients to remove parentheses on both sides of the equation.
4p−10+3p=−3p+24+2p−104p - 10 + 3p = -3p + 24 + 2p - 10
To simplify the equation, all terms inside parentheses must be multiplied by their outer coefficients. Note that 23×6p=4p\frac{2}{3} \times 6p = 4p and 23×−15=−10\frac{2}{3} \times -15 = -10.
2
Combine like terms on each side of the equation independently.
7p−10=−p+147p - 10 = -p + 14
Combining variable terms and constant terms on the same side simplifies the equation structure before isolating the variable.
3
Move all variable terms to one side of the equation.
8p−10=148p - 10 = 14
Adding pp to both sides cancels the −p-p on the right side, grouping all unknowns on the left.
4
Isolate the variable by moving constants to the other side and dividing.
8p=24⇒p=38p = 24 \Rightarrow p = 3
Adding 10 to both sides moves the constant, and dividing both sides by the coefficient 8 solves for pp.

Anahtar Kavram

Solving complex one-variable linear equations requiring distribution and combining like terms.
Soru 4929Soru →

A renewable energy company is installing a chain-link fence around a newly constructed rectangular solar farm. The solar farm measures 120120 yards in length and 8585 yards in width. The fence will enclose the entire perimeter, with the exception of two access gates that each measure 55 yards wide. If the fencing material costs $22\$22 per linear yard, what is the total cost of the fencing material in dollars?

Cevabı ve açıklamayı göster

Cevap: 8800

Cevap

The total cost of the fencing material is 8800 dollars.
The total perimeter of the solar farm is 410410 yards (120+120+85+85120 + 120 + 85 + 85). Subtracting the 1010 yards for the two 55-yard gates leaves 400400 yards of fencing to be installed. Multiplying 400400 yards by the $22\$22 per yard cost gives a total of $8800\$8800.

Adım Adım Çözüm

1
Calculate the total perimeter of the rectangular solar farm.
P=2×(120+85)=2×205=410P = 2 \times (120 + 85) = 2 \times 205 = 410 yards.
The perimeter formula for a rectangle is P=2(L+W)P = 2(L + W), which represents the total boundary length before any openings are considered.
2
Determine the total width of the two access gates.
2×5=102 \times 5 = 10 yards.
Since there are two gates and each is 55 yards wide, their combined width must be subtracted from the total perimeter.
3
Calculate the actual length of fencing needed.
410−10=400410 - 10 = 400 yards.
The fencing covers the perimeter minus the space left open for the access gates.
4
Calculate the total cost of the fencing material.
400×22=8800400 \times 22 = 8800.
Multiplying the total yards of fencing by the cost per yard gives the total cost in dollars.

Anahtar Kavram

Calculating the perimeter of a rectangle, subtracting openings, and finding the total cost based on a unit price.
Soru 4930Soru →

Consider the following algebraic equation, where x≠0x \neq 0:

(3x4)29xk=x5 \frac{(3x^4)^2}{9x^k} = x^5

What is the value of the exponent kk?

Cevabı ve açıklamayı göster

Cevap: 3; k=3; k = 3; three

Cevap

3
First, the numerator (3x4)2(3x^4)^2 simplifies to 9x89x^8. Substituting this back yields 9x89xk=x5\frac{9x^8}{9x^k} = x^5. The coefficients cancel out, and applying the quotient rule to the variables gives x8−k=x5x^{8-k} = x^5. By equating the exponents, we get 8−k=58 - k = 5, which means k=3k = 3.

Adım Adım Çözüm

1
Apply the power of a product rule to the numerator.
(3x4)2=32⋅(x4)2=9x8(3x^4)^2 = 3^2 \cdot (x^4)^2 = 9x^8
When raising a product to a power, you must raise every factor inside the parentheses to that power.
2
Substitute the simplified numerator back into the equation and simplify the numerical coefficients.
9x89xk=x8xk\frac{9x^8}{9x^k} = \frac{x^8}{x^k}
The coefficients 99\frac{9}{9} cancel out to 1, making it easier to isolate the variable base.
3
Apply the quotient rule for exponents on the left side of the equation.
x8xk=x8−k\frac{x^8}{x^k} = x^{8-k}
When dividing exponential expressions with the same base, you subtract the exponent in the denominator from the exponent in the numerator.
4
Set the resulting exponent equal to the exponent on the right side of the original equation and solve for kk.
8−k=5⇒k=38 - k = 5 \Rightarrow k = 3
Since the bases on both sides of the equation are the same (xx), their exponents must be equal.

Anahtar Kavram

Operations with Variable Exponents
Soru 4931Soru →

A homeowner is planning to enclose a rectangular residential lot with a new privacy fence. The lot has a length of 2020 yards and a width of 4545 feet. The fencing material costs $12.50\$12.50 per linear foot. If the homeowner needs to leave a 66-foot opening for a driveway gate, what will be the total cost of the fencing materials?

Cevabı ve açıklamayı göster

Cevap: $2,550\$2,550

Cevap

The total cost of the fencing materials is $2,550\$2,550.
The correct answer is found by first converting all units to feet. The length of 2020 yards is equal to 6060 feet. The total perimeter of the rectangular lot is 2×(60+45)=2102 \times (60 + 45) = 210 feet. Subtracting the 66-foot gap for the gate leaves 204204 linear feet of fencing required. Multiplying this by the cost of $12.50\$12.50 per foot yields a total cost of $2,550\$2,550.

Adım Adım Çözüm

1
Convert the length of the lot from yards to feet.
20 yards×3=60 feet20 \text{ yards} \times 3 = 60 \text{ feet}
All measurements must be in the same unit (feet) because the fencing cost is given per linear foot.
2
Calculate the total perimeter of the rectangular lot.
P=2×(60+45)=2×105=210 feetP = 2 \times (60 + 45) = 2 \times 105 = 210 \text{ feet}
The fencing will go around the entire boundary of the rectangular lot.
3
Subtract the length of the driveway gate from the total perimeter.
210−6=204 feet210 - 6 = 204 \text{ feet}
Fencing material is not needed for the space where the gate will be installed.
4
Calculate the total cost by multiplying the fencing length by the cost per foot.
204×$12.50=$2,550204 \times \$12.50 = \$2,550
The total cost is the required length in feet multiplied by the price per linear foot.

Anahtar Kavram

Calculating the perimeter of a rectangle with mixed units and adjusting for an opening.
Soru 4932Soru →

If x=−2x = -2 and y=−3y = -3, what is the value of the expression 2x2−5y2x^2 - 5y?

Cevabı ve açıklamayı göster

Cevap: 23

Cevap

23
The expression is evaluated by first substituting the values and applying the exponent to the base, giving (−2)2=4(-2)^2 = 4. Then, perform the multiplication to get 2(4)=82(4) = 8 and 5(−3)=−155(-3) = -15. Finally, subtracting negative 15 from 8 is the same as adding 15, resulting in 8+15=238 + 15 = 23.

Adım Adım Çözüm

1
Substitute the given values for xx and yy into the expression.
2(−2)2−5(−3)2(-2)^2 - 5(-3)
To evaluate the expression, each variable must be replaced with its corresponding numerical value.
2
Evaluate the exponent.
2(4)−5(−3)2(4) - 5(-3)
According to the order of operations (PEMDAS), exponents must be resolved before multiplication. The square of any non-zero real number is positive, so (−2)2=4(-2)^2 = 4.
3
Perform the multiplication operations.
8−(−15)8 - (-15)
Multiply 22 by 44 to get 88, and 55 by −3-3 to get −15-15.
4
Perform the subtraction.
8+15=238 + 15 = 23
Subtracting a negative number is mathematically identical to adding its positive counterpart.

Anahtar Kavram

Evaluating expressions using substitution and the order of operations.
Soru 4933Soru →

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?

Cevabı ve açıklamayı göster

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

Cevap

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}.

Adım Adım Çözüm

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.

Anahtar Kavram

Applying exponent rules (power, product, and quotient) with numerical coefficients and negative integers.
Soru 4934Soru →

What is the value of the expression 12−2∣−4+1∣12 - 2|-4 + 1|?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6
The expression is evaluated by first simplifying the terms inside the absolute value to get |-3|, which equals positive 3. Multiplying this by the coefficient 2 gives 6. Finally, subtracting 6 from 12 yields the correct answer of 6.

Adım Adım Çözüm

1
Simplify the expression inside the absolute value bars.
−4+1=−3-4 + 1 = -3
According to the order of operations, operations inside absolute value bars (which act as grouping symbols) must be performed first.
2
Evaluate the absolute value of the result.
∣−3∣=3|-3| = 3
The absolute value of a number represents its distance from zero, which is always positive.
3
Multiply the result by the coefficient outside the absolute value.
2×3=62 \times 3 = 6
Multiplication must be performed before subtraction according to the order of operations (PEMDAS).
4
Subtract the product from the initial value.
12−6=612 - 6 = 6
This completes the evaluation of the expression to find the final numerical value.

Anahtar Kavram

Absolute Value and Order of Operations
Soru 4935Soru →

Arrange the following fractions and mixed numbers in ascending order (from smallest to largest).

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct ascending order is 78\frac{7}{8}, 1141 \frac{1}{4}, 32\frac{3}{2}, and 1341 \frac{3}{4}.
The proper fraction 78\frac{7}{8} is the only value less than 1, placing it first. Converting 32\frac{3}{2} to 1121 \frac{1}{2} (or 1.51.5) allows easy comparison with 1141 \frac{1}{4} (1.25) and 1341 \frac{3}{4} (1.75), confirming the correct ascending sequence.

Adım Adım Çözüm

1
Identify values less than 1.
78\frac{7}{8} is the only proper fraction (less than 1), making it the smallest.
Proper fractions are always less than mixed numbers or improper fractions.
2
Convert the improper fraction to a mixed number to make comparison easier.
32=112\frac{3}{2} = 1 \frac{1}{2}.
Matching the format of the other numbers allows for a direct comparison of the fractional parts.
3
Find a common denominator for the fractional parts of the values greater than 1 (1141 \frac{1}{4}, 1121 \frac{1}{2}, and 1341 \frac{3}{4}).
The common denominator is 4. 1121 \frac{1}{2} becomes 1241 \frac{2}{4}.
Common denominators allow you to directly compare the numerators.
4
Order the mixed numbers by their fractional parts.
1141 \frac{1}{4} is smaller than 1241 \frac{2}{4} (which is 32\frac{3}{2}), and 1241 \frac{2}{4} is smaller than 1341 \frac{3}{4}.
Since the whole numbers are identical, the fractions determine the order.

Anahtar Kavram

Ordering mixed numbers and improper fractions
Soru 4936Soru →

A network engineer is configuring a load balancer to distribute traffic across two servers. The equation below models the traffic distribution, where tt represents the number of active connections:

5(2t−4)−3t=4(t+2)−(t−8)5(2t - 4) - 3t = 4(t + 2) - (t - 8)

Calculate the required number of active connections, tt, to successfully balance the server loads.

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

9
By successfully applying the distributive property on both sides, combining the variable and constant terms, and isolating the variable, the equation perfectly balances when the variable is exactly 9.

Adım Adım Çözüm

1
Distribute the coefficients and negative signs across the parentheses on both sides of the equation.
10t−20−3t=4t+8−t+810t - 20 - 3t = 4t + 8 - t + 8
Distribution removes the parentheses, allowing individual terms to be combined.
2
Combine like terms on each side of the equation.
7t−20=3t+167t - 20 = 3t + 16
Simplifying each side individually makes it easier to balance the equation.
3
Subtract 3t3t from both sides of the equation.
4t−20=164t - 20 = 16
This groups all the variable terms on one side of the equation.
4
Add 20 to both sides and divide by 4.
4t=36  ⟹  t=94t = 36 \implies t = 9
Isolating the variable determines the final numerical solution.

Anahtar Kavram

Solving multi-step linear equations involving the distributive property and combining like terms.
Soru 4937Soru →

What is the product of the binomials (2x−1)(2x - 1) and (x+4)(x + 4)?

Cevabı ve açıklamayı göster

Cevap: 2x2+7x−42x^2 + 7x - 4

Cevap

The correct expression is 2x2+7x−42x^2 + 7x - 4.
By applying the FOIL method, we multiply the First terms to get 2x22x^2, the Outer terms to get 8x8x, the Inner terms to get −x-x, and the Last terms to get −4-4. Combining the like terms 8x8x and −x-x results in the simplified polynomial 2x2+7x−42x^2 + 7x - 4.

Adım Adım Çözüm

1
Multiply the First terms.
2x×x=2x22x \times x = 2x^2
This is the 'F' step in the FOIL method.
2
Multiply the Outer terms.
2x×4=8x2x \times 4 = 8x
This is the 'O' step in the FOIL method.
3
Multiply the Inner terms.
−1×x=−x-1 \times x = -x
This is the 'I' step in the FOIL method.
4
Multiply the Last terms.
−1×4=−4-1 \times 4 = -4
This is the 'L' step in the FOIL method.
5
Combine the resulting terms.
2x2+8x−x−4=2x2+7x−42x^2 + 8x - x - 4 = 2x^2 + 7x - 4
Combine the like middle terms (8x8x and −x-x) to simplify the final expression.

Anahtar Kavram

Multiplying binomials using the distributive property (FOIL method) and combining like terms.
Soru 4938Soru →

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.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Öğeler

(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}

Eşleşmeler

Cevabı ve açıklamayı göster

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Operations with Variable Exponents
Soru 4939Soru →

A botanist has 120120 tulip bulbs, 168168 daffodil bulbs, and 216216 crocus bulbs. She wants to plant them in separate rows such that each row contains only one type of flower, and every row has the exact same number of bulbs. What is the greatest number of bulbs she can plant in a single row?

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Cevap: 24

Cevap

24
The greatest number of bulbs per row must perfectly divide the total count of each flower type without remainders. Therefore, we calculate the Greatest Common Factor (GCF) of 120, 168, and 216. The prime factorizations reveal the common factors are 232^3 and 313^1, which multiply to 24.

Adım Adım Çözüm

1
Identify the mathematical operation required by the scenario.
The problem asks for the greatest number that divides 120, 168, and 216 evenly. This means finding the Greatest Common Factor (GCF).
Each row must have the same number of bulbs and only one flower type, meaning the row size must be a common divisor of all three total quantities.
2
Find the prime factorization of 120, 168, and 216.
120=23×3×5120 = 2^3 \times 3 \times 5, 168=23×3×7168 = 2^3 \times 3 \times 7, and 216=23×33216 = 2^3 \times 3^3.
Prime factorization is a systematic method to determine the common factors of multiple large numbers.
3
Identify the common prime factors and select their lowest powers.
The common prime factors are 2 and 3. Their lowest powers across the three numbers are 232^3 and 313^1.
The GCF is formed by multiplying the lowest powers of all prime factors that appear in every number's factorization.
4
Calculate the GCF.
23×3=8×3=242^3 \times 3 = 8 \times 3 = 24.
Multiplying the selected prime factors yields the greatest possible number of bulbs per row.

Anahtar Kavram

Greatest Common Factor (GCF)
Soru 4940Soru →

Evaluate the following mathematical expressions involving decimals. What is the correct sequence of these expressions when ordered from least to greatest value?

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from least to greatest is: 0.15×40−14.4÷1.20.15 \times 40 - 14.4 \div 1.2, followed by 10.5−3.2×2.510.5 - 3.2 \times 2.5, then (4.2−1.8)÷0.6+0.5(4.2 - 1.8) \div 0.6 + 0.5, and finally 2.4×0.5+1.8÷0.32.4 \times 0.5 + 1.8 \div 0.3.
Evaluating each expression using the correct order of operations yields final decimal values of 7.27.2, 2.52.5, −6.0-6.0, and 4.54.5. Arranging these values numerically from least to greatest provides the correct structural sequence.

Adım Adım Çözüm

1
Evaluate the first expression: 2.4×0.5+1.8÷0.32.4 \times 0.5 + 1.8 \div 0.3.
1.2+6=7.21.2 + 6 = 7.2
According to the order of operations, multiplication and division are performed left-to-right before addition.
2
Evaluate the second expression: 10.5−3.2×2.510.5 - 3.2 \times 2.5.
10.5−8.0=2.510.5 - 8.0 = 2.5
Multiplication must be performed before subtraction.
3
Evaluate the third expression: 0.15×40−14.4÷1.20.15 \times 40 - 14.4 \div 1.2.
6.0−12=−6.06.0 - 12 = -6.0
Perform multiplication and division first. Then subtract the larger value from the smaller value, resulting in a negative number.
4
Evaluate the fourth expression: (4.2−1.8)÷0.6+0.5(4.2 - 1.8) \div 0.6 + 0.5.
2.4÷0.6+0.5=4+0.5=4.52.4 \div 0.6 + 0.5 = 4 + 0.5 = 4.5
Operations inside parentheses take highest priority, followed by division, and finally addition.
5
Order the evaluated results from least to greatest.
−6.0<2.5<4.5<7.2-6.0 < 2.5 < 4.5 < 7.2
Arranging the numerical values based on their algebraic magnitude provides the final sequence.

Anahtar Kavram

Order of Operations with Decimals

Alternatif Yöntem

Convert all terminating decimals to fractions (e.g., 0.5=120.5 = \frac{1}{2}, 0.3=3100.3 = \frac{3}{10}) before performing the operations to avoid decimal placement and alignment errors.
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