Power of a Product: Variables in the exponent of the power

Solving the equationVariable in the exponent of the powerSame base and different indicatorUsing multiple rulesCombination of different basesCalculating powers with negative exponentsVariable in the base of the powerInverse formulaNumber of termsApplying the formula
Question 1

Solve the following exercise:

\( (4\times9\times11)^a \)

Question 2

\( (2\cdot4\cdot8)^{a+3}= \)

Question 3

Simplify:

\( (2\cdot3\cdot7\cdot9)^{ab+3} \)

Question 4

Insert the corresponding expression:

\( \left(9\times7\times4\right)^a= \)

Question 5

Insert the corresponding expression:

\( \)\( \left(7\times8\right)^{b+a}= \)

Examples with solutions for Power of a Product: Variables in the exponent of the power

Exercise #1

Solve the following exercise:

(4×9×11)a (4\times9\times11)^a

Video Solution

Step-by-Step Solution

We use the power law for a multiplication between parentheses:

(z⋅t)n=zn⋅tn (z\cdot t)^n=z^n\cdot t^n

That is, a power applied to a multiplication between parentheses is applied to each term when the parentheses are opened,

We apply it in the problem:

(4⋅9⋅11)a=4a⋅9a⋅11a (4\cdot9\cdot11)^a=4^a\cdot9^a\cdot11^a

Therefore, the correct answer is option b.

Note:

From the power property formula mentioned, we can understand that it works not only with two terms of the multiplication between parentheses, but also valid with a multiplication between multiple terms in parentheses. As we can see in this problem.

Answer

4a×9a×11a 4^a\times9^a\times11^a

Exercise #2

(2⋅4⋅8)a+3= (2\cdot4\cdot8)^{a+3}=

Video Solution

Step-by-Step Solution

Let's begin by using the distributing exponents rule (An exponent outside of a parentheses needs to be distributed across all the numbers and variables within the parentheses)

(x⋅y)n=xn⋅yn (x\cdot y)^n=x^n\cdot y^n We first apply this rule to the given problem:

(2⋅4⋅8)a+3=2a+34a+38a+3 (2\cdot4\cdot8)^{a+3}= 2^{a+3}4^{a+3}8^{a+3} When then we apply the power to each of the terms of the product inside the parentheses separately and maintain the multiplication.

The correct answer is option d.

Answer

2a+34a+38a+3 2^{a+3}4^{a+3}8^{a+3}

Exercise #3

Simplify:

(2⋅3⋅7⋅9)ab+3 (2\cdot3\cdot7\cdot9)^{ab+3}

Video Solution

Step-by-Step Solution

We begin by using the distributive law of exponents.

(x⋅y)n=xn⋅yn (x\cdot y)^n=x^n\cdot y^n

We apply this property to the given problem :

(2⋅3⋅7⋅9)ab+3=2ab+33ab+37ab+39ab+3 (2\cdot3\cdot7\cdot9)^{ab+3} =2^{ab+3}3^{ab+3}7^{ab+3}9^{ab+3}

When we apply the power of parentheses to each of the terms of the product inside the parentheses separately and maintain the multiplication.

Therefore, the correct answer is option a.

Answer

2ab+33ab+37ab+39ab+3 2^{ab+3}3^{ab+3}7^{ab+3}9^{ab+3}

Exercise #4

Insert the corresponding expression:

(9×7×4)a= \left(9\times7\times4\right)^a=

Video Solution

Answer

9a×7a×4a 9^a\times7^a\times4^a

Exercise #5

Insert the corresponding expression:

(7×8)b+a= \left(7\times8\right)^{b+a}=

Video Solution

Answer

7b+a×8b+a 7^{b+a}\times8^{b+a}

Question 1

Insert the corresponding expression:

\( \left(2\times8\right)^{2y+2}= \)

Question 2

Insert the corresponding expression:

\( \left(3\times4\right)^{3x+1}= \)

Question 3

Insert the corresponding expression:

\( \left(3\times6\times4\right)^{2a}= \)

Question 4

Insert the corresponding expression:

\( \left(4\times6\right)^{b+2}= \)

Question 5

Insert the corresponding expression:

\( \left(4\times6\right)^b= \)

Exercise #6

Insert the corresponding expression:

(2×8)2y+2= \left(2\times8\right)^{2y+2}=

Video Solution

Answer

22y+2×82y+8 2^{2y+2}\times8^{2y+8}

Exercise #7

Insert the corresponding expression:

(3×4)3x+1= \left(3\times4\right)^{3x+1}=

Video Solution

Answer

33x+1×43x+1 3^{3x+1}\times4^{3x+1}

Exercise #8

Insert the corresponding expression:

(3×6×4)2a= \left(3\times6\times4\right)^{2a}=

Video Solution

Answer

32a×62a×42a 3^{2a}\times6^{2a}\times4^{2a}

Exercise #9

Insert the corresponding expression:

(4×6)b+2= \left(4\times6\right)^{b+2}=

Video Solution

Answer

4b+2×6b+2 4^{b+2}\times6^{b+2}

Exercise #10

Insert the corresponding expression:

(4×6)b= \left(4\times6\right)^b=

Video Solution

Answer

4b×6b 4^b\times6^b

Question 1

Insert the corresponding expression:

\( \left(5\times3\right)^{6x}= \)

Question 2

Insert the corresponding expression:

\( \left(6\times7\times10\right)^{y+4+a}= \)

Question 3

Insert the corresponding expression:

\( \left(7\times5\times2\right)^{y+4}= \)

Question 4

Insert the corresponding expression:

\( \left(7\times5\times2\right)^y= \)

Question 5

Insert the corresponding expression:

\( \left(8\times2\right)^x= \)

Exercise #11

Insert the corresponding expression:

(5×3)6x= \left(5\times3\right)^{6x}=

Video Solution

Answer

56x×36x 5^{6x}\times3^{6x}

Exercise #12

Insert the corresponding expression:

(6×7×10)y+4+a= \left(6\times7\times10\right)^{y+4+a}=

Video Solution

Answer

6y+4+a×7y+4+a×10y+4+a 6^{y+4+a}\times7^{y+4+a}\times10^{y+4+a}

Exercise #13

Insert the corresponding expression:

(7×5×2)y+4= \left(7\times5\times2\right)^{y+4}=

Video Solution

Answer

7y+4×5y+4×2y+4 7^{y+4}\times5^{y+4}\times2^{y+4}

Exercise #14

Insert the corresponding expression:

(7×5×2)y= \left(7\times5\times2\right)^y=

Video Solution

Answer

5y×7y×5y 5^y\times7^y\times5^y

Exercise #15

Insert the corresponding expression:

(8×2)x= \left(8\times2\right)^x=

Video Solution

Answer

8x×2x 8^x\times2^x

Question 1

Insert the corresponding expression:

\( \left(9\times2\right)^{3x}= \)

Question 2

Insert the corresponding expression:

\( 3^x\times7^x\times5^x= \)

Question 3

Insert the corresponding expression:

\( 4^a\times8^a\times2^a= \)

Question 4

Insert the corresponding expression:

\( 5^{y+1}\times3^{y+1}= \)

Question 5

Insert the corresponding expression:

\( 5^y\times3^y= \)

Exercise #16

Insert the corresponding expression:

(9×2)3x= \left(9\times2\right)^{3x}=

Video Solution

Answer

93x×23x 9^{3x}\times2^{3x}

Exercise #17

Insert the corresponding expression:

3x×7x×5x= 3^x\times7^x\times5^x=

Video Solution

Answer

(3×7×5)x \left(3\times7\times5\right)^x

Exercise #18

Insert the corresponding expression:

4a×8a×2a= 4^a\times8^a\times2^a=

Video Solution

Answer

(4×8×2)a \left(4\times8\times2\right)^a

Exercise #19

Insert the corresponding expression:

5y+1×3y+1= 5^{y+1}\times3^{y+1}=

Video Solution

Answer

(5×3)y+1 \left(5\times3\right)^{y+1}

Exercise #20

Insert the corresponding expression:

5y×3y= 5^y\times3^y=

Video Solution

Answer

(5×3)y \left(5\times3\right)^y

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