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

We use the formula

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

Therefore, we obtain:

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

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 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:

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