Power of a multiplication - Examples, Exercises and Solutions

When finding an expression with multiplication or an exercise that has only multiplication operations inside a parenthesis and the wholes expression is raised to a certain exponent, we can take the exponent and apply it to each of the terms of the expression or exercise.
We must not forget to keep the multiplication signs between the terms.
Property formula:
(a×b)n=an×bn (a\times b)^n=a^n\times b^n
This property also pertains to algebraic expressions.

Suggested Topics to Practice in Advance

  1. Power of a Power

Practice Power of a multiplication

Exercise #1

(5x3)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

153x3 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

(ab8)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

a2b282 a^2\cdot b^2\cdot8^2

Exercise #4

(a56y)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

a5305y5 a^5\cdot30^5\cdot y^5

Exercise #5

(4×7×3)2= (4\times7\times3)^2=

Video Solution

Step-by-Step Solution

We use the power law for multiplication within parentheses:

(xy)n=xnyn (x\cdot y)^n=x^n\cdot y^n We apply it to the problem:

(473)2=427232 (4\cdot7\cdot3)^2=4^2\cdot7^2\cdot3^2 Therefore, the correct answer is option a.

Note:

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

Answer

42×72×32 4^2\times7^2\times3^2

Exercise #1

(3×4×5)4= (3\times4\times5)^4=

Video Solution

Step-by-Step Solution

We use the power law for multiplication within parentheses:

(xy)n=xnyn (x\cdot y)^n=x^n\cdot y^n We apply it to the problem:

(345)4=344454 (3\cdot4\cdot5)^4=3^4\cdot4^4\cdot5^4 Therefore, the correct answer is option b.

Note:

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

Answer

344454 3^44^45^4

Exercise #2

(9×2×5)3= (9\times2\times5)^3=

Video Solution

Step-by-Step Solution

We use the power property for a power that is applied to parentheses in which terms are multiplied:

(xy)n=xnyn (x\cdot y)^n=x^n\cdot y^n We apply the property in the problem:

(925)3=932353 (9\cdot2\cdot5)^3=9^3\cdot2^3\cdot5^3 When we apply the power within parentheses to the product of the terms to each term of the product separately and maintain the multiplication,

Therefore, the correct answer is option B.

Answer

93×23×53 9^3\times2^3\times5^3

Exercise #3

(2×8×7)2= (2\times8\times7)^2=

Video Solution

Step-by-Step Solution

We use the power property for the product inside parentheses:

(zt)n=zntn (z\cdot t)^n=z^n\cdot t^n That is, the power applied to a product inside parentheses is applied to each term of it when the parentheses are opened,

We apply the property to the problem:

(287)2=228272 (2\cdot8\cdot7)^2=2^2\cdot8^2\cdot7^2 Therefore, the correct answer is option d.

Note:

From the formula of the power property inside parentheses mentioned above, it might seem that it refers only to two terms of the product inside parentheses, but in reality, it is also valid for the power over a multiplication of many terms inside parentheses, as was done in this problem and in other problems.

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

12,544 12,544

Exercise #4

(x43)3= (x\cdot4\cdot3)^3=

Video Solution

Step-by-Step Solution

Use the power property for a power that is applied to parentheses in which terms are multiplied:

(xy)n=xnyn (x\cdot y)^n=x^n\cdot y^n We apply the property in the problem:

(x43)3=x34333 (x\cdot4\cdot3)^3= x^3\cdot4^3\cdot3^3 When we apply the power to parentheses to the product of the terms, we apply the power to each term of the product separately and keep the product,

Therefore, the correct answer is option C.

Answer

x34333 x^3\cdot4^3\cdot3^3

Exercise #5

(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 #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:

(zt)n=zntn (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:

(4911)a=4a9a11a (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

4a9a11a 4^a9^a11^a

Exercise #2

(y×7×3)4= (y\times7\times3)^4=

Video Solution

Step-by-Step Solution

We use the power law for multiplication within parentheses:

(xy)n=xnyn (x\cdot y)^n=x^n\cdot y^n We apply it in the problem:

(y73)4=y47434 (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 #3

(7463)4=? (7\cdot4\cdot6\cdot3)^4= \text{?}

Video Solution

Step-by-Step Solution

We use the power property for an exponent that is applied to a set parentheses in which the terms are multiplied:

(xy)n=xnyn (x\cdot y)^n=x^n\cdot y^n We apply the law in the problem:

(7463)4=74446434 (7\cdot4\cdot6\cdot3)^4=7^4\cdot4^4\cdot6^4\cdot3^4 When we apply the exponent to a parentheses with multiplication, we apply the exponent to each term of the multiplication separately, and we keep the multiplication between them.

Therefore, the correct answer is option a.

Answer

74446434 7^4\cdot4^4\cdot6^4\cdot3^4

Exercise #4

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

Video Solution

Step-by-Step Solution

We use the power property for the product inside parentheses:

(zt)n=zntn (z\cdot t)^n=z^n\cdot t^n That is, the power applied to a product inside parentheses is applied to each term of it when the parentheses are opened,

We apply the property to the problem:

(3246)4=34244464 (3\cdot2\cdot4\cdot6)^{-4}=3^{-4}\cdot2^{-4}\cdot4^{-4}\cdot6^{-4} Therefore, the correct answer is option d.

Note:

From the formula of the power property inside parentheses mentioned above, it can be understood that it refers only to two terms of the product inside parentheses, but in reality, it is also valid for the power over a multiplication of many terms inside parentheses, as was done in this problem and in other problems.

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

34×24×44×64 3^{-4}\times2^{-4}\times4^{-4}\times6^{-4}

Exercise #5

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

Video Solution

Step-by-Step Solution

Use the power property for an exponent that applies to parentheses in which terms are multiplied:

(xy)n=xnyn (x\cdot y)^n=x^n\cdot y^n We apply this property to the expression of the problem:

(248)a+3=2a+34a+38a+3 (2\cdot4\cdot8)^{a+3}= 2^{a+3}4^{a+3}8^{a+3} When we apply the power within parentheses 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}

Topics learned in later sections

  1. Power of a Quotient
  2. Multiplying Exponents with the Same Base
  3. Division of Exponents with the Same Base
  4. Rules of Exponentiation
  5. Combining Powers and Roots