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

Solve the exercise:

$(a^5)^7=$

We use the formula:

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

and therefore we obtain:

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

$a^{35}$

$(a^4)^6=$

We use the formula

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

Therefore, we obtain:

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

$a^{24}$

$((y^6)^8)^9=$

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.

$y^{432}$

$((a^2)^3)^{\frac{1}{4}}=$

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.

$a^{1.5}$

$((b^3)^6)^2=$

We use the formula

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

Therefore, we obtain:

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

$b^{36}$

Solve the exercise:

$(x^2\times3)^2=$

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.

$9x^4$

$(g\times a\times x)^4+(4^a)^x=$

$g^4a^4x^4+4^{ax}$