Question

what exponent rules and properties can you apply to prove that \\(\sqrt4{27}=27^{\frac{1}{4}}\\)? explain how you rewrite each part of the equation. be sure to name the rules of exponents and properties of equality that you use in each step. type your answer in the box. use the \\(\sqrt{x}\\) button to enter math expressions or equations.

Explanation:

Step1: Recall the definition of radicals and exponents

The radical form $\sqrt[n]{a}$ (where $n$ is the index of the radical and $a$ is the radicand) can be rewritten using the exponent rule that relates radicals and exponents. The rule is $\sqrt[n]{a}=a^{\frac{1}{n}}$, which comes from the property of exponents and roots: if we consider that taking the $n$th root of a number is the same as raising that number to the power of $\frac{1}{n}$.

In the given equation, we have the 4th root of 27, which is $\sqrt[4]{27}$. Using the rule $\sqrt[n]{a}=a^{\frac{1}{n}}$ with $n = 4$ and $a=27$, we can rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$.

Answer:

To prove that $\boldsymbol{\sqrt[4]{27}=27^{\frac{1}{4}}}$, we use the exponent rule for radicals: $\boldsymbol{\sqrt[n]{a}=a^{\frac{1}{n}}}$ (where $n$ is the index of the radical and $a$ is the radicand).

• For $\sqrt[4]{27}$, the index of the radical $n = 4$ and the radicand $a = 27$. Applying the rule $\sqrt[n]{a}=a^{\frac{1}{n}}$, we substitute $n = 4$ and $a = 27$:

$\sqrt[4]{27}=27^{\frac{1}{4}}$.

This shows the two expressions are equivalent by the definition of how radicals and fractional exponents relate.