Algebra Toolkit

By the definition of square root, \(\sqrt{a}\) is a number whose square is \(a\text{.}\)

1. \(\displaystyle \left(\sqrt{7}\right)\left(\sqrt{7}\right)=7\)
2. \(\displaystyle \sqrt{n}(\sqrt{n})=n\)

By the definition of cube root, \(\sqrt[3]{a}\) is a number whose cube is \(a\text{.}\)

1. \(\displaystyle \left(\sqrt[3]{5}\right)^3=5\)
2. \(\displaystyle \left(\sqrt[3]{4}\right)\left(\sqrt[3]{4}\right)\left(\sqrt[3]{4}\right)=4\)

1. \(\displaystyle \dfrac{3}{\sqrt{3}} = \dfrac{\sqrt{3}\sqrt{3}}{\sqrt{3}} = \sqrt{3}\)
2. \(\displaystyle \dfrac{p}{\sqrt{p}} = \dfrac{\sqrt{p}\sqrt{p}}{\sqrt{p}} = \sqrt{p}\)

1. Irrational: \(~\sqrt{6}~\) is not the quotient of two integers.

2. Rational: \(~\dfrac{-5}{3}~\) is the quotient of two integers.

3. Rational: \(~\sqrt{16} = 4~\) is an integer.

4. Irrational: \(~\sqrt{\dfrac{5}{9}} = \dfrac{\sqrt{5}}{3}~\text{,}\) but \(\sqrt{5}\) is irrational.

Use a calculator to evaluate.

1. Enter \(~5~ \boxed{\sqrt{}}~3\) ENTER and round to three decimal places: \(~8.660 \)
2. Enter (-) \(2\) \(\boxed{\sqrt{}}~\) \(21\) ) ÷ \(3\) ENTER and round to three decimal places: \(~-3.055\)
3. Enter \(2\) + MATH 4 \(5\) ENTER and round to three decimal places: \(~3.710\)

1. We start by simplifying the numerator.

   \begin{align*} \dfrac{4-\sqrt{64}}{2} \amp = \dfrac{4-8}{2} \amp\amp \blert{\text{Evaluate the radical, then subtract.}}\\ \amp = \dfrac{-4}{2}=-2 \amp\amp \blert{\text{Reduce the fraction.}} \end{align*}

2. We start by simplifyng what's inside parentheses.

   \begin{align*} -2\amp\left(3\sqrt{16} -\sqrt{3(27)}\right) \amp\amp \blert{\text{Evaluate the radicals.}}\\ \amp = -2(3 \cdot 4-\sqrt{81}) \amp\amp \blert{\text{Simplify inside the parentheses.}}\\ \amp = -2(12-9)=-6 \end{align*}

3. We start by simplifying the radicand.

   \begin{align*} 6-3\sqrt[3]{27-7(5)} \amp = 6-3\sqrt[3]{27-35} \amp\amp \blert{\text{Subtract under the radical.}}\\ \amp = 6-3\sqrt[3]{-8} \amp\amp \blert{\text{Evaluate the radical.}}\\ \amp =6-3(-2)=12 \end{align*}

1. Do not start with "\(8-2\)"! Evaluate \(\sqrt{2}\) first, then multiply by 2, and subtract the result from 8. Once the numerator is simplified, divide by 4.

   On a calculator, enter

   \(\qquad\qquad\) ( \(8\) - \(2~\boxed{\sqrt{}}\) ) ) ÷ \(4\) ENTER

   and round to two decimal places: \(~1.29\)

2. Evaluate the cube root, multiply by 6, then add the result to 2.

   On a calculator, enter

   \(\qquad\qquad 2\) + \(6\) MATH 4 - \(25\) ENTER

   and round to two decimal places: \(~{-15.54}\)