Question

Please help!! 10 points

Answer 1

Answer:

We conclude that:

$\:\sqrt[3]{200k^{15}}=2k^5\sqrt[3]{25}$

Hence, option B is correct.

Step-by-step explanation:

Given the expression

$\sqrt[3]{200k^{15}}$

Apply radical rule:

$\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b},\:\quad \mathrm{\:assuming\:}a\ge 0,\:b\ge 0$

so the expression becomes

$\sqrt[3]{200k^{15}}=\sqrt[3]{200}\sqrt[3]{k^{15}}$

first solving

$\sqrt[3]{k^{15}}$

Apply radical rule: $\sqrt[n]{a^m}=a^{\frac{m}{n}},\:\quad \mathrm{\:assuming\:}a\ge 0$

$\sqrt[3]{k^{15}}=k^{\frac{15}{3}}=k^5$

then solving

$\sqrt[3]{200}$

prime factorization:  200:  2³ · 5²

$=\sqrt[3]{2^3\cdot \:5^2}$

Apply radical rule:

$\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b},\:\quad \mathrm{\:assuming\:}a\ge 0,\:b\ge 0$

$=\sqrt[3]{2^3}\sqrt[3]{5^2}$

Apply radical rule:  

$\sqrt[n]{a^n}=a,\:\quad \:a\ge 0$

so

$=2\sqrt[3]{5^2}$

Thus, the main expression becomes

$\sqrt[3]{200k^{15}}=\sqrt[3]{200}\sqrt[3]{k^{15}}$

              $=2k^5\sqrt[3]{25}$

Therefore, we conclude that:

$\:\sqrt[3]{200k^{15}}=2k^5\sqrt[3]{25}$

Hence, option B is correct.