Chapter 1.2, Problem 33E

To determine

To simplify: the given expression.

Answer to Problem 33E

  $\left(x^2+4\right)\sqrt{x}$ .

Explanation of Solution

Given information:

Given expression

  $\sqrt{16x}+\sqrt{x^5}$

Calculation:

Consider the given expression.

  $\sqrt{16x}+\sqrt{x^5}$

The expression can be simplified as shown:

Here, we have $n$ th roots, in this case, because they are square roots, $n=2$ .

  $\sqrt[2]{16x}+\sqrt[2]{x^5}$

Let’s start by using the first property of $n$ th roots, which states

Rewrite the first term using this.

  $\sqrt[2]{16}\sqrt[2]{x}+\sqrt[2]{x^5}$

This simplifies to

Now, we can use the definition of the $n$ th roots, which is

  $a^{\frac{1}{n}}=\sqrt[n]{a}$

And rewrite the second term. The first term can just be left as a square root for now.

  $\sqrt[4]{x}+{\left(x^5\right)}^{\frac{1}{2}}$

The third law of exponents, which states

  ${\left(a^m\right)}^n=a^{mn}$

Allows us to multiply the exponents of the second term.

  $\sqrt[4]{x}+x^{\frac{5}{2}}$

The first law of exponents, which states

  $a^m\cdot a^n=a^{m+n}$ ,

Helps us split up the second term $\left(2+\frac{1}{2}=\frac{5}{2}\right)$ .

  $\begin{array}{l}\sqrt[4]{x}+x^2\cdot x^{\frac{1}{2}}\\ \sqrt[4]{x}+x^2\cdot \sqrt{x}\end{array}$

Factor the square root.

  $\left(x^2+4\right)\sqrt{x}$

The simplified expression is $\left(x^2+4\right)\sqrt{x}$ .