Question

1 write the radical expression \\(\frac{8}{\sqrt7{x^{15}}}\\) in exponential form. \\(\boldsymbol{\text{a}}\\) \\(8x^{\frac{7}{15}}\\) \\(\boldsymbol{\text{b}}\\) \\(8x^{-\frac{7}{15}}\\) \\(\boldsymbol{\text{c}}\\) \\(8x^{\frac{15}{7}}\\) \\(\boldsymbol{\text{d}}\\) \\(8x^{-\frac{15}{7}}\\)

Explanation:

Step1: Recall radical to exponent rule

The formula for converting a radical \(\sqrt[n]{a^m}\) to exponential form is \(a^{\frac{m}{n}}\), and \(\frac{1}{\sqrt[n]{a^m}}=a^{-\frac{m}{n}}\).

Step2: Apply the rule to the denominator

For \(\frac{8}{\sqrt[7]{x^{15}}}\), first handle \(\sqrt[7]{x^{15}}\). Using the rule, \(\sqrt[7]{x^{15}} = x^{\frac{15}{7}}\), so \(\frac{1}{\sqrt[7]{x^{15}}}=x^{-\frac{15}{7}}\).

Step3: Multiply by the numerator

The numerator is 8, so the entire expression \(\frac{8}{\sqrt[7]{x^{15}}}\) in exponential form is \(8\times x^{-\frac{15}{7}}\), which is \(8x^{-\frac{15}{7}}\) (wait, no, wait: Wait, the exponent for the denominator: \(\sqrt[7]{x^{15}}\) is \(x^{15/7}\), so reciprocal is \(x^{- 15/7}\)? Wait, no, wait the options have -15/7? Wait the options: B is \(8x^{-\frac{7}{15}}\)? No, wait no, let me check again. Wait, no, the radical is 7th root of \(x^{15}\), so \(\sqrt[7]{x^{15}}=x^{15/7}\), so \(\frac{1}{\sqrt[7]{x^{15}}}=x^{-15/7}\)? But the options: D is \(8x^{-15/7}\)? Wait the original problem: the radical is \(\sqrt[7]{x^{15}}\)? Wait the user's image: the radical is \(\sqrt[7]{x^{15}}\)? Wait the options: D is \(8x^{-15/7}\)? Wait let me re - check the steps.

Wait, the formula is \(\frac{1}{\sqrt[n]{a^m}}=a^{-\frac{m}{n}}\). So here, \(n = 7\), \(m = 15\), so \(\frac{1}{\sqrt[7]{x^{15}}}=x^{-\frac{15}{7}}\). Then multiply by 8, so the expression is \(8x^{-\frac{15}{7}}\), which is option D. Wait, maybe I misread the options. Let me check the options again:

A: \(8x^{\frac{7}{15}}\)

B: \(8x^{-\frac{7}{15}}\)

C: \(8x^{\frac{15}{7}}\)

D: \(8x^{-\frac{15}{7}}\)

Yes, so step by step:

1. Recall that \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\) and \(\frac{1}{a^b}=a^{-b}\).

1. For \(\sqrt[7]{x^{15}}\), using the radical - to - exponent rule, we have \(\sqrt[7]{x^{15}}=x^{\frac{15}{7}}\).

1. Then \(\frac{1}{\sqrt[7]{x^{15}}}=\frac{1}{x^{\frac{15}{7}}}=x^{-\frac{15}{7}}\) (by the negative exponent rule \(a^{-n}=\frac{1}{a^n}\)).

1. Multiply by the numerator 8, so \(\frac{8}{\sqrt[7]{x^{15}}}=8\times x^{-\frac{15}{7}} = 8x^{-\frac{15}{7}}\).

Answer:

D. \(8x^{-\frac{15}{7}}\)