Question

1. rewrite the following radical expressions using rational exponents.

a. $sqrt3{x^2}$
b. $sqrt4{x^3y}$

Explanation:

Part (a)

Step1: Recall the radical to exponent rule

The formula for converting a radical to a rational exponent is $\sqrt[n]{a^m}=a^{\frac{m}{n}}$. For $\sqrt[3]{x^2}$, here $n = 3$ (the index of the cube root) and $m=2$ (the exponent of the base $x$ inside the radical).

Step2: Apply the rule

Using the rule $\sqrt[n]{a^m}=a^{\frac{m}{n}}$, substitute $n = 3$, $m = 2$ and $a=x$ into the formula. So we get $x^{\frac{2}{3}}$.

Step1: Recall the radical to exponent rule for nested radicals

First, for a nested radical like $\sqrt[4]{x^3y}$, we can use the rule $\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}$ (product rule of radicals) and $\sqrt[n]{a^m}=a^{\frac{m}{n}}$. Let's break down $\sqrt[4]{x^3y}$ as $\sqrt[4]{x^3}\cdot\sqrt[4]{y}$.

Step2: Apply the radical - exponent conversion rule to each part

For $\sqrt[4]{x^3}$, using $\sqrt[n]{a^m}=a^{\frac{m}{n}}$, with $n = 4$, $m = 3$ and $a = x$, we get $x^{\frac{3}{4}}$. For $\sqrt[4]{y}$, with $n=4$, $m = 1$ (since $y=y^1$) and $a = y$, we get $y^{\frac{1}{4}}$. Then, by the product rule of exponents $a^m\cdot a^n=a^{m + n}$ (but here we are multiplying two different bases, so we just write them as a product of their exponential forms). So $\sqrt[4]{x^3y}=x^{\frac{3}{4}}y^{\frac{1}{4}}$.

Answer:

$x^{\frac{2}{3}}$

Part (b)