Power of a Product: Variable in the base of the power

Solving the equationVariable in the exponent of the powerSame base and different indicatorUsing multiple rulesCombination of different basesVariables in the exponent of the powerCalculating powers with negative exponentsInverse formulaNumber of termsApplying the formula
Question 1

\( (5\cdot x\cdot3)^3= \)

Question 2

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

Question 3

\( (x\cdot4\cdot3)^3= \)

Question 4

\( (a\cdot b\cdot8)^2= \)

Question 5

\( (a\cdot5\cdot6\cdot y)^5= \)

Examples with solutions for Power of a Product: Variable in the base of the power

Exercise #1

(5⋅x⋅3)3= (5\cdot x\cdot3)^3=

Video Solution

Step-by-Step Solution

We use the formula:

(a×b)n=anbn (a\times b)^n=a^nb^n

(5×x×3)3=(15x)3 (5\times x\times3)^3=(15x)^3

(15x)3=(15×x)3 (15x)^3=(15\times x)^3

153x3 15^3x^3

Answer

153⋅x3 15^3\cdot x^3

Exercise #2

(y×x×3)5= (y\times x\times3)^5=

Video Solution

Step-by-Step Solution

We use the formula:

(a×b)n=anbn (a\times b)^n=a^nb^n

(y×x×3)5=y5x535 (y\times x\times3)^5=y^5x^53^5

Answer

y5×x5×35 y^5\times x^5\times3^5

Exercise #3

(x⋅4⋅3)3= (x\cdot4\cdot3)^3=

Video Solution

Step-by-Step Solution

Let us begin by using the law of exponents for a power that is applied to parentheses in which terms are multiplied:

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

(x⋅4⋅3)3=x3⋅43⋅33 (x\cdot4\cdot3)^3= x^3\cdot4^3\cdot3^3 When we apply the power to the product of the terms within parentheses, we apply the power to each term of the product separately and keep the product,

Therefore, the correct answer is option C.

Answer

x3⋅43⋅33 x^3\cdot4^3\cdot3^3

Exercise #4

(a⋅b⋅8)2= (a\cdot b\cdot8)^2=

Video Solution

Step-by-Step Solution

We use the formula

(a×b)x=axbx (a\times b)^x=a^xb^x

Therefore, we obtain:

a2b282 a^2b^28^2

Answer

a2⋅b2⋅82 a^2\cdot b^2\cdot8^2

Exercise #5

(a⋅5⋅6⋅y)5= (a\cdot5\cdot6\cdot y)^5=

Video Solution

Step-by-Step Solution

We use the formula:

(a×b)x=axbx (a\times b)^x=a^xb^x

Therefore, we obtain:

(a×5×6×y)5=(a×30×y)5 (a\times5\times6\times y)^5=(a\times30\times y)^5

a5305y5 a^530^5y^5

Answer

a5⋅305⋅y5 a^5\cdot30^5\cdot y^5

Question 1

\( (a\times b\times c\times4)^7= \)

Question 2

\( (y\times7\times3)^4= \)

Question 3

Insert the corresponding expression:

\( \left(y\times x\right)^2= \)

Question 4

Insert the corresponding expression:

\( \left(y\times a\right)^5= \)

Question 5

Insert the corresponding expression:

\( \left(a\times b\right)^3= \)

Exercise #6

(a×b×c×4)7= (a\times b\times c\times4)^7=

Video Solution

Step-by-Step Solution

We use the formula:

(a×b)x=axbx (a\times b)^x=a^xb^x

Therefore, we obtain:

a7b7c747 a^7b^7c^74^7

Answer

a7×b7×c7×47 a^7\times b^7\times c^7\times4^7

Exercise #7

(y×7×3)4= (y\times7\times3)^4=

Video Solution

Step-by-Step Solution

We use the power law for multiplication within parentheses:

(x⋅y)n=xn⋅yn (x\cdot y)^n=x^n\cdot y^n We apply it in the problem:

(y⋅7⋅3)4=y4⋅74⋅34 (y\cdot7\cdot3)^4=y^4\cdot7^4\cdot3^4 Therefore, the correct answer is option a.

Note:

From the formula of the power property mentioned above, we can understand that it applies not only to two terms within parentheses, but also for multiple terms within parentheses.

Answer

y4×74×34 y^4\times7^4\times3^4

Exercise #8

Insert the corresponding expression:

(y×x)2= \left(y\times x\right)^2=

Video Solution

Answer

y2×x2 y^2\times x^2

Exercise #9

Insert the corresponding expression:

(y×a)5= \left(y\times a\right)^5=

Video Solution

Answer

y5×a5 y^5\times a^5

Exercise #10

Insert the corresponding expression:

(a×b)3= \left(a\times b\right)^3=

Video Solution

Answer

a3×b3 a^3\times b^3

Question 1

Insert the corresponding expression:

\( \left(5\times b\times a\right)^4= \)

Question 2

Insert the corresponding expression:

\( \left(b\times9\times a\right)^6= \)

Question 3

Insert the corresponding expression:

\( \left(c\times b\times a\right)^2= \)

Question 4

Insert the corresponding expression:

\( \left(b\times z\times a\right)^5= \)

Question 5

Insert the corresponding expression:

\( \)\( \left(2\times x\right)^2= \)

Exercise #11

Insert the corresponding expression:

(5×b×a)4= \left(5\times b\times a\right)^4=

Video Solution

Answer

54×b4×a4 5^4\times b^4\times a^4

Exercise #12

Insert the corresponding expression:

(b×9×a)6= \left(b\times9\times a\right)^6=

Video Solution

Answer

b6×96×a6 b^6\times9^6\times a^6

Exercise #13

Insert the corresponding expression:

(c×b×a)2= \left(c\times b\times a\right)^2=

Video Solution

Answer

a'+b' are correct

Exercise #14

Insert the corresponding expression:

(b×z×a)5= \left(b\times z\times a\right)^5=

Video Solution

Answer

a5×b5×z5 a^5\times b^5\times z^5

Exercise #15

Insert the corresponding expression:

(2×x)2= \left(2\times x\right)^2=

Video Solution

Answer

22×x2 2^2\times x^2

Question 1

Insert the following expression:

\( \left(y\times3\right)^2= \)

Question 2

Insert the corresponding expression:

\( \left(a\times3\right)^3= \)

Question 3

Insert the corresponding expression:

\( \left(6\times b\right)^4= \)

Question 4

Insert the corresponding expression:

\( \left(7\times4\times a\right)^5= \)

Question 5

Insert the corresponding expression:

\( x^7\times9^7\times y^7= \)

Exercise #16

Insert the following expression:

(y×3)2= \left(y\times3\right)^2=

Video Solution

Answer

y2×32 y^2\times3^2

Exercise #17

Insert the corresponding expression:

(a×3)3= \left(a\times3\right)^3=

Video Solution

Answer

a3×33 a^3\times3^3

Exercise #18

Insert the corresponding expression:

(6×b)4= \left(6\times b\right)^4=

Video Solution

Answer

64×b4 6^4\times b^4

Exercise #19

Insert the corresponding expression:

(7×4×a)5= \left(7\times4\times a\right)^5=

Video Solution

Answer

75×45×a5 7^5\times4^5\times a^5

Exercise #20

Insert the corresponding expression:

x7×97×y7= x^7\times9^7\times y^7=

Video Solution

Answer

(x×9×y)7 \left(x\times9\times y\right)^7