Power of a Power - Examples, Exercises and Solutions

To solve the exercise we use the power property:$(a^n)^m=a^{n\cdot m}$

We use the property with our exercise and solve:

$(3^5)^4=3^{5\times4}=3^{20}$

We use the formula:

$(a^n)^m=a^{n\times m}$

Therefore, we obtain:

$6^{2\times13}=6^{26}$

We use the formula:

$(a^m)^n=a^{m\times n}$

and therefore we obtain:

$(a^5)^7=a^{5\times7}=a^{35}$

We use the formula:

$(a^m)^n=a^{m\times n}$

$(4^2)^3+(g^3)^4=4^{2\times3}+g^{3\times4}=4^6+g^{12}$

We use the formula

$(a^m)^n=a^{m\times n}$

Therefore, we obtain:

$a^{4\times6}=a^{24}$

We use the power property of a negative exponent:

$a^{-n}=\frac{1}{a^n}$We will rewrite the fraction in parentheses as a negative power:

$\frac{1}{7}=7^{-1}$Let's return to the problem, where we had:

$\bigg( \big( \frac{1}{7}\big)^{-1}\bigg)^4=\big((7^{-1})^{-1} \big)^4$We continue and use the power property of an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$And we apply it in the problem:

$\big((7^{-1})^{-1} \big)^4 =(7^{-1\cdot-1})^4=(7^1)^4=7^{1\cdot4}=7^4$Therefore, the correct answer is option c

We use the power rule of distributing exponents.

$(a^m)^n=a^{m\cdot n}$We apply it in the problem:

$\big((y^6)^8\big)^9=(y^{6\cdot8})^9=y^{6\cdot8\cdot9}=y^{432}$When we use the aforementioned rule twice, the first time for the inner parentheses in the first stage and the second time for the remaining parentheses in the second stage, in the last stage we calculate the result of the multiplication in the power exponent.

Therefore, the correct answer is option b.

We have an exponent raised to another exponent with a multiplication between parentheses:

$(z\cdot t)^n=z^n\cdot t^n$This says that in a case where a power is applied to a multiplication between parentheses,the power is applied to each term of the multiplication when the parentheses are opened,

We apply it in the problem:

$(3x^2)^2=3^2(x^2)^2$With the second term of the multiplication we proceed carefully, since it is already in a power (that's why we use parentheses). The term will be raised using the power law for an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$and we apply it in the problem:

$3^2(x^2)^2=9x^{2\cdot2}=9x^4$In the first step we raise the number to the power, and in the second step we multiply the exponent.

Therefore, the correct answer is option a.

We use the formula:

$(a^m)^n=a^{m\times n}$

$(2^2)^3+(3^3)^4+(9^2)^6=2^{2\times3}+3^{3\times4}+9^{2\times6}=2^6+3^{12}+9^{12}$

Using the law of powers for an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$We apply it in the problem:

$(4^x)^y=4^{xy}$Therefore, the correct answer is option a.

We use the power rule for exponents.

$(a^m)^n=a^{m\cdot n}$We apply it to the problem:

$\big((a^2)^3\big)^{\frac{1}{4}}=(a^{2\cdot3})^{\frac{1}{4}}=a^{2\cdot3\cdot\frac{1}{4}}=a^{\frac{6}{4}}=a^{\frac{3}{2}}$When we use the previously mentioned rule twice, the first time for the inner parentheses in the first stage and the second time for the remaining parentheses in the second stage, in the third stage we calculate the result of the multiplication in the exponent. While remembering that multiplying by a fraction is actually doubling the numerator of the fraction and, finally, in the last stage we simplify the fraction we obtained in the exponent.

Now remember that -

$\frac{3}{2}=1\frac{1}{2}=1.5$

Therefore, the correct answer is option a.

We use the formula

$(a^m)^n=a^{m\times n}$

Therefore, we obtain:

$((b^3)^6)^2=(b^{3\times6})^2=(b^{18})^2=b^{18\times2}=b^{36}$

Using the power rule for an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$We apply the rule to the given problem:

$((14^{3x})^{2y})^{5a}=(14^{3x})^{2y\cdot5a}=14^{3x\cdot2y\cdot5a}=14^{30xya}$In the first step we applied the aforementioned power rule and removed the outer parentheses. In the next step we again applied the power rule and removed the remaining parentheses.

In the final step we simplified the resulting expression,

Therefore, through the rule of substitution (which is applied to the exponent of the power in the obtained expression) it can be concluded that the correct answer is answer D.

We use the power rule for an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$We apply this rule to the given problem:

$((3^9)^{4x})^{5y}= (3^9)^{4x\cdot 5y} =3^{9\cdot4x\cdot 5y}=3^{180xy}$In the first step we applied the previously mentioned power rule and removed the outer parentheses. In the next step we applied the power rule once again and removed the remaining parentheses. In the final step we simplified the resulting expression.

Therefore, the correct answer is option b.

We'll use the power rule for powers:

$(a^m)^n=a^{m\cdot n}$We'll apply this rule to the expression in the problem:

$((4x)^{3y})^2= (4x)^{3y\cdot2}=(4x)^{6y}$When in the first stage we applied the mentioned power rule and eliminated the outer parentheses, in the next stage we simplified the resulting expression,

Next, we'll recall the power rule for powers that applies to parentheses containing a product of terms:

$(a\cdot b)^n=a^n\cdot b^n$We'll apply this rule to the expression we got in the last stage:

$(4x)^{6y} =4^{6y}\cdot x^{6y}$When we applied the power to the parentheses to each term of the product inside the parentheses.

Therefore, the correct answer is answer D.