Power of a multiplication - Examples, Exercises and Solutions

$(2\times8\times7)^2=$

We use the power property for the product inside parentheses:

$(z\cdot t)^n=z^n\cdot t^n$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:

$(2\cdot8\cdot7)^2=2^2\cdot8^2\cdot7^2$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.).

$12,544$

$(9\times2\times5)^3=$

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

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

$(9\cdot2\cdot5)^3=9^3\cdot2^3\cdot5^3$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.

$9^3\times2^3\times5^3$

$(3\times4\times5)^4=$

We use the power law for multiplication within parentheses:

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

$(3\cdot4\cdot5)^4=3^4\cdot4^4\cdot5^4$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.

$3^44^45^4$

$(4\times7\times3)^2=$

We use the power law for multiplication within parentheses:

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

$(4\cdot7\cdot3)^2=4^2\cdot7^2\cdot3^2$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.

$4^2\times7^2\times3^2$

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

We use the power property for the product inside parentheses:

$(z\cdot t)^n=z^n\cdot t^n$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:

$(3\cdot2\cdot4\cdot6)^{-4}=3^{-4}\cdot2^{-4}\cdot4^{-4}\cdot6^{-4}$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.).

$3^{-4}\times2^{-4}\times4^{-4}\times6^{-4}$

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

We use the formula:

$(a\times b)^n=a^nb^n$

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

$(15x)^3=(15\times x)^3$

$15^3x^3$

$15^3\cdot x^3$

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

We use the formula:

$(a\times b)^n=a^nb^n$

$(y\times x\times3)^5=y^5x^53^5$

$y^5\times x^5\times3^5$

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

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

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

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

$x^3\cdot4^3\cdot3^3$

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

We use the formula

$(a\times b)^x=a^xb^x$

Therefore, we obtain:

$a^2b^28^2$

$a^2\cdot b^2\cdot8^2$

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

We use the formula:

$(a\times b)^x=a^xb^x$

Therefore, we obtain:

$(a\times5\times6\times y)^5=(a\times30\times y)^5$

$a^530^5y^5$

$a^5\cdot30^5\cdot y^5$

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

We use the formula:

$(a\times b)^x=a^xb^x$

Therefore, we obtain:

$a^7b^7c^74^7$

$a^7\times b^7\times c^7\times4^7$

Solve the following exercise:

$(4\times9\times11)^a$

We use the power law for a multiplication between parentheses:

$(z\cdot t)^n=z^n\cdot t^n$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:

$(4\cdot9\cdot11)^a=4^a\cdot9^a\cdot11^a$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.

$4^a9^a11^a$

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

We use the power law for multiplication within parentheses:

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

$(y\cdot7\cdot3)^4=y^4\cdot7^4\cdot3^4$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.

$y^4\times7^4\times3^4$

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

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

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

$(7\cdot4\cdot6\cdot3)^4=7^4\cdot4^4\cdot6^4\cdot3^4$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.

$7^4\cdot4^4\cdot6^4\cdot3^4$

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

We use the power property for the product within parentheses:

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

We apply the property to the problem:

$(8\cdot9\cdot5\cdot3)^{-2}=8^{-2}\cdot9^{-2}\cdot5^{-2}\cdot3^{-2}$Therefore, the correct answer is option c.

Note:

From the formula of the power property within parentheses mentioned above, it can be understood that it refers only to two terms of the product within parentheses, but in reality, it is also valid for the power over a multiplication of many terms within 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 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.).

$8^{-2}\times9^{-2}\times5^{-2}\times3^{-2}$