Question 1

Which of the following is the correct definition of the principal nth root of a real number $$a$$, where $$n \geq 2 is an $$integer?

Accepted Answer

$$n\sqrt{a} = b$$ means $$b^n = a$$ and $$b \geq 0 if$$ $$n is $$even, $$a \geq 0$$; $$b is $$any real number if $$n is $$odd$

Question 2

Simplify $$\sqrt[3]{-8}.$$

Accepted Answer

$$-2$$

Question 3

Which property of radicals allows you to write $$\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$$ for $$a, b \geq 0$$?

Accepted Answer

Product property of radicals

Question 4

Use a calculator to approximate $$\sqrt[4]{18}. $$Which value is closest?

Accepted Answer

$$2.06$$

Question 5

Simplify $$\sqrt{50}.$$

Accepted Answer

$$5\sqrt{2}$$

Question 6

Combine like radicals: $$4\sqrt{20} - 5\sqrt{5}.$$

Accepted Answer

$$3\sqrt{5}$$

Question 7

Rationalize the denominator: $$\frac{2}{\sqrt{3}}.$$

Accepted Answer

$$\frac{2\sqrt{3}}{3}$$

Question 8

Rationalize the numerator: $$\frac{\sqrt{x+h} - \sqrt{x}}{h}$$, $$h 
eq 0.$$

Accepted Answer

$$\frac{1}{\sqrt{x+h} + \sqrt{x}}$$

Question 9

Express $$9^{1/2} as a $$radical and simplify.

Accepted Answer

$$\sqrt{9} = 3$$

Question 10

Simplify the expression $$\$$left(x^{2/3}$$ y$$^{1/2}$$\right) \$$left(x^{-1/2}$$ y$$^{2/3}$$\right)$$, assuming $$x, y \u003e 0.$$

Accepted Answer

$$x^{1/6}$$ $$y^{7/6}$$

Question 11

Write $$54^{1/3} as a $$radical and simplify as much as possible.

Accepted Answer

$$3\sqrt{2}$$

Question 12

Which of the following is equivalent to $$a^{m/n}$$ for $$a \u003e 0$$, $$n \geq 2$$?

Accepted Answer

$$\sqrt[n]{a^m}$$

Question 13

Simplify $$\$$left(8x^3$$y^2$$\$$right)^{1/3}$$, assuming $$x, y \u003e 0$.

Accepted Answer

$$2x y$$^{2/3}$$

Question 14

Write the following as a single quotient with only positive exponents: $$x^{1/3}$$ + $$x^{2/3}.$$

Accepted Answer

$$x^{1/3}(1$$ + $$x^{1/3}$$)$$

Question 15

Factor and simplify: $$x^{1/2}$$ + $$2x^{3/2}.$$

Accepted Answer

$$x^{1/2}(1 + 2x$$)$$

Question 16

Which of the following is the correct way to rationalize the denominator of $$\frac{1}{\sqrt{2} - 3}$$?

Accepted Answer

Multiply numerator and denominator by $$\sqrt{2} + 3$$

Question 17

A chemist needs to dilute a solution and calculates the amount using $$\sqrt[4]{x}$$, where $$x is $$the concentration. If $$x = 81$$, what is the result?

Accepted Answer

$$3$$

Question 18

Simplify $$\left(\frac{y}{x}\$$right)^{2/3}$$, assuming x$$, y \u003e 0$$.

Accepted Answer

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

Question 19

A medical researcher models the growth of bacteria with $$N = 100 \cdot t$$^{1/2}$$, where t$ is time in hours. What is N$ when t = 16$?

Accepted Answer

$$400$$

Chapter R.8: nth Roots and Rational Exponents – College Algebra Study Notes

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