化简 $(16y^3)^{\frac{3}{2}}$。 $5^k = 125\sqrt[3]{5}\sqrt{5}$ 利用指数定律求 $k$ 的值。
写出你的计算过程。 答 k = ...

1. Apply the power to each factor and simplify.

$(16y^3)^{\frac{3}{2}}=16^{\frac{3}{2}}(y^3)^{\frac{3}{2}}=\left(\sqrt{16}\right)^3y^{3\times\frac{3}{2}}=4^3y^{\frac{9}{2}}=64y^{\frac{9}{2}}$

1. Express all terms on the right-hand side as powers of 5.

$125=5^3,\quad \sqrt[3]{5}=5^{\frac13},\quad \sqrt{5}=5^{\frac12}$

So
$5^k=5^3\times 5^{\frac13}\times 5^{\frac12}.$

2. Use the law of indices for multiplying powers with the same base.

$5^k=5^{3+\frac13+\frac12}.$

3. Add the fractional indices and equate powers of the same base.

$3+\frac13+\frac12=3+\frac{2}{6}+\frac{3}{6}=3+\frac{5}{6}=\frac{23}{6}.$

Since the bases are the same,
$k=\frac{23}{6}.$