Power of a Product: Calculating powers with negative exponents

Solving the equationVariable in the exponent of the powerSame base and different indicatorUsing multiple rulesCombination of different basesVariables in the exponent of the powerVariable in the base of the powerInverse formulaNumber of termsApplying the formula
Question 1

\( (3\times2\times4\times6)^{-4}= \)

Question 2

\( (8\times9\times5\times3)^{-2}= \)

Question 3

Simplify:

\( (5\cdot12\cdot4\cdot6)^{a+3bx} \)

Question 4

Insert the corresponding expression:

\( \left(5\times8\right)^{-5}= \)

Question 5

Insert the corresponding expression:

\( \left(2\times9\times6\right)^{-7}= \)

Examples with solutions for Power of a Product: Calculating powers with negative exponents

Exercise #1

(3×2×4×6)−4= (3\times2\times4\times6)^{-4}=

Video Solution

Step-by-Step Solution

We begin by using the power rule for parentheses.

(z⋅t)n=zn⋅tn (z\cdot t)^n=z^n\cdot t^n That is, the power applied to a product inside parentheses is applied to each of the terms within when the parentheses are opened,

We apply the above rule to the given problem:

(3⋅2⋅4⋅6)−4=3−4⋅2−4⋅4−4⋅6−4 (3\cdot2\cdot4\cdot6)^{-4}=3^{-4}\cdot2^{-4}\cdot4^{-4}\cdot6^{-4} Therefore, the correct answer is option d.

Note:

According to the formula of the power property inside parentheses mentioned above, it might seem as though it refers to only two terms of the product inside of the parentheses, but in reality, it is also valid for the power over a multiplication of many terms inside parentheses, as was seen above.

A good exercise is to demonstrate that if the previous property is valid for a power over a product of two terms inside parentheses (as formulated above), then it is also valid for a power over several terms of the product inside parentheses (for example - three terms, etc.).

Answer

3−4×2−4×4−4×6−4 3^{-4}\times2^{-4}\times4^{-4}\times6^{-4}

Exercise #2

(8×9×5×3)−2= (8\times9\times5\times3)^{-2}=

Video Solution

Step-by-Step Solution

We begin by applying the power rule to the products within the parentheses:

(z⋅t)n=zn⋅tn (z\cdot t)^n=z^n\cdot t^n That is, the power applied to a product within parentheses is applied to each of the terms when the parentheses are opened,

We apply the rule to the given problem:

(8⋅9⋅5⋅3)−2=8−2⋅9−2⋅5−2⋅3−2 (8\cdot9\cdot5\cdot3)^{-2}=8^{-2}\cdot9^{-2}\cdot5^{-2}\cdot3^{-2} Therefore, the correct answer is option c.

Note:

Whilst it could be understood that the above power rule applies only to two terms of the product within parentheses, in reality, it is also valid for the power over a multiplication of multiple terms within parentheses, as was seen in the above problem.

A good exercise is to demonstrate that if the previous property is valid for a power over a product of two terms within parentheses (as formulated above), then it is also valid for a power over several terms of the product within parentheses (for example - three terms, etc.).

Answer

8−2×9−2×5−2×3−2 8^{-2}\times9^{-2}\times5^{-2}\times3^{-2}

Exercise #3

Simplify:

(5⋅12⋅4⋅6)a+3bx (5\cdot12\cdot4\cdot6)^{a+3bx}

Video Solution

Step-by-Step Solution

Use the power property for a power in parentheses where there is a multiplication of its terms:

(x⋅y)n=xn⋅yn (x\cdot y)^n=x^n\cdot y^n We apply this law to the problem expression:

(5⋅12⋅4⋅6)a+3bx=5a+3bx12a+3bx4a+3bx6a+3bx (5\cdot12\cdot4\cdot6)^{a+3bx}=5^{a+3bx}12^{a+3bx}4^{a+3bx}6^{a+3bx} When we apply a power to parentheses where its terms are multiplied, we do it separately and keep the multiplication.

Therefore, the correct answer is option d.

Answer

5a+3bx12a+3bx4a+3bx6a+3bx 5^{a+3bx}12^{a+3bx}4^{a+3bx}6^{a+3bx}

Exercise #4

Insert the corresponding expression:

(5×8)−5= \left(5\times8\right)^{-5}=

Video Solution

Answer

1(5×8)5 \frac{1}{\left(5\times8\right)^5}

Exercise #5

Insert the corresponding expression:

(2×9×6)−7= \left(2\times9\times6\right)^{-7}=

Video Solution

Answer

1(2×9×6)7 \frac{1}{\left(2\times9\times6\right)^7}

Question 1

Insert the corresponding expression:

\( \left(9\times7\times8\right)^{-8}= \)

Question 2

Insert the corresponding expression:

\( \left(10\times7\times9\right)^{-5}= \)

Question 3

Insert the corresponding expression:

\( \left(4\times7\right)^{-2}= \)

Question 4

Insert the corresponding expression:

\( \left(6\times5\right)^{-3}= \)

Question 5

Insert the corresponding expression:

\( \left(2\times7\times3\right)^{-6}= \)

Exercise #6

Insert the corresponding expression:

(9×7×8)−8= \left(9\times7\times8\right)^{-8}=

Video Solution

Answer

1(9×7×8)8 \frac{1}{\left(9\times7\times8\right)^8}

Exercise #7

Insert the corresponding expression:

(10×7×9)−5= \left(10\times7\times9\right)^{-5}=

Video Solution

Answer

1(10×7×9)5 \frac{1}{\left(10\times7\times9\right)^5}

Exercise #8

Insert the corresponding expression:

(4×7)−2= \left(4\times7\right)^{-2}=

Video Solution

Answer

142×72 \frac{1}{4^2\times7^2}

Exercise #9

Insert the corresponding expression:

(6×5)−3= \left(6\times5\right)^{-3}=

Video Solution

Answer

163×53 \frac{1}{6^3\times5^3}

Exercise #10

Insert the corresponding expression:

(2×7×3)−6= \left(2\times7\times3\right)^{-6}=

Video Solution

Answer

A+B are correct

Question 1

Insert the corresponding expression:

\( \left(5\times6\times8\right)^{-9}= \)

Question 2

Insert the corresponding expression:

\( \left(12\times7\times9\right)^{-5}= \)

Question 3

Insert the corresponding expression:

\( \left(4\times5\right)^{-2}= \)

Question 4

Insert the corresponding expression:

\( \left(9\times2\right)^{-6}= \)

Question 5

Insert the corresponding expression:

\( \left(2\times7\times3\right)^{-3}= \)

Exercise #11

Insert the corresponding expression:

(5×6×8)−9= \left(5\times6\times8\right)^{-9}=

Video Solution

Answer

159×69×89 \frac{1}{5^9\times6^9\times8^9}

Exercise #12

Insert the corresponding expression:

(12×7×9)−5= \left(12\times7\times9\right)^{-5}=

Video Solution

Answer

1125×75×95 \frac{1}{12^5\times7^5\times9^5}

Exercise #13

Insert the corresponding expression:

(4×5)−2= \left(4\times5\right)^{-2}=

Video Solution

Answer

4−2×5−2 4^{-2}\times5^{-2}

Exercise #14

Insert the corresponding expression:

(9×2)−6= \left(9\times2\right)^{-6}=

Video Solution

Answer

9−6×2−6 9^{-6}\times2^{-6}

Exercise #15

Insert the corresponding expression:

(2×7×3)−3= \left(2\times7\times3\right)^{-3}=

Video Solution

Answer

2−3×7−3×3−3 2^{-3}\times7^{-3}\times3^{-3}

Question 1

Insert the corresponding expression:

\( \left(9\times7\times8\right)^{-8}= \)

Question 2

Insert the corresponding expression:

\( \left(3\times7\right)^{-4}= \)

Question 3

Insert the corresponding expression:

\( \frac{1}{\left(2\times7\times8\right)^9}= \)

Question 4

Insert the corresponding expression:

\( \frac{1}{\left(10\times7\right)^5}= \)

Question 5

Insert the corresponding expression:

\( \frac{1}{8^{-7}\times9^{-7}\times5^{-7}}= \)

Exercise #16

Insert the corresponding expression:

(9×7×8)−8= \left(9\times7\times8\right)^{-8}=

Video Solution

Answer

9−8×7−8×8−8 9^{-8}\times7^{-8}\times8^{-8}

Exercise #17

Insert the corresponding expression:

(3×7)−4= \left(3\times7\right)^{-4}=

Video Solution

Answer

1(3×7)4 \frac{1}{\left(3\times7\right)^4}

Exercise #18

Insert the corresponding expression:

1(2×7×8)9= \frac{1}{\left(2\times7\times8\right)^9}=

Video Solution

Answer

(2×7×8)−9 \left(2\times7\times8\right)^{-9}

Exercise #19

Insert the corresponding expression:

1(10×7)5= \frac{1}{\left(10\times7\right)^5}=

Video Solution

Answer

(10×5)−5 \left(10\times5\right)^{-5}

Exercise #20

Insert the corresponding expression:

18−7×9−7×5−7= \frac{1}{8^{-7}\times9^{-7}\times5^{-7}}=

Video Solution

Answer

(8×9×5)7 \left(8\times9\times5\right)^7