Question

Simplify the expression

An explanation will be greatly appreciated


`\frac{1}{(\sqrt[3]{8p^(6))^(4) } }`

Answer 1

Explanation:

`\frac{1}{(\sqrt[3]{(8p^(6))^(4) } } \\ \\ = \frac{1}{(\sqrt[3]{ ({2}^(3) p^(6))^(4) } } \\ \\ = \frac{1}{(\sqrt[3]{ ({2}^(3 * 4) p^(6 * 4)) } }\\ \\ = \frac{1}{(\sqrt[3]{ ({2}^(12) * p^(24)) } } \\ \\ = \frac{1}{({ {2}^{12 * (1)/(3) } * p^{24 * (1)/(3) }) } } \\ \\ = \frac{1}{({ {2}^(4) * p^(8 )) } } \\ \\ \purple{ \boxed{ \bold{ \therefore \: \frac{1}{(\sqrt[3]{(8p^(6))^(4) } } = \frac{1}{{16 p^(8 ) } }}}}`

Answer 2

Nice job inputting the expression.

The cube root is a 1/3 power. The 4 in the denominator is a -4 power in the numerator. When we have powers of powers we multiply them all together. When we have a product to a power we have to raise each factor to the power.

We get to choose whether we want a fraction at the end or negative exponents. Because of the constant 16 in the denominator I chose fraction.

`\frac{1}{( \sqrt[3]{8p^6})^4} = ( \sqrt[3]{8p^6})^(-4) = ( (8p^6)^(\frac 1 3))^(-4) =(8^(\frac 1 3))^(-4) p^((6(-4)/3)) = 2^(-4) p^(-8) = (1)/(16p^8)`

Answer: 1/(16p⁸)