# Simplify expressions with square roots

Given below are exponent questions for grade 9 math students.
All the questions are provided with detailed solution.

To solve the questions, you should have basic knowledge of exponent and algebra rules.

Question 01
Simplify the below expression.

\mathtt{\sqrt[3]{4} \ \times \sqrt[3]{16}}

Solution
Using the following formula;
\mathtt{\sqrt{a} .\sqrt{b} =\sqrt{ab}}\\\ \\

\mathtt{\Longrightarrow \ \sqrt[3]{4} \ \times \sqrt[3]{16}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt[3]{4\times 16}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt[3]{64}}

\mathtt{\sqrt[3]{64} \ =\ 2\ \times 2\ =\ 4}

Hence, 4 is the right answer.

Question 02
Solve the below expression.

\mathtt{\frac{\sqrt[4]{1250}}{\sqrt[4]{2}}}

Solution
Using the following formula;
\mathtt{\frac{\sqrt{a}}{\sqrt{b}} =\sqrt{\frac{a}{b}}} \\\ \\

\mathtt{\Longrightarrow \ \frac{\sqrt[4]{1250}}{\sqrt[4]{2}}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt[4]{1250\ \div \ 2}}\\\ \\ \mathtt{\Longrightarrow \sqrt[4]{625}}

\mathtt{\sqrt[4]{625} \ =\ 5}

Hence, 5 is the right answer.

Question 03
Simplify the below expression;
\mathtt{\left( 3+\sqrt{7}\right)\left( 4-\sqrt{2}\right)}

Solution
Solving the given expression;

\mathtt{\Longrightarrow \left( 3+\sqrt{7}\right)\left( 4-\sqrt{2}\right)}\\\ \\ \mathtt{\Longrightarrow 3\times 4-3\sqrt{2} +4\sqrt{7} -\sqrt{7}\sqrt{2}}\\\ \\ \mathtt{\Longrightarrow \ 12-3\sqrt{2} +4\sqrt{7} -\sqrt{7\times 2}}\\\ \\ \mathtt{\Longrightarrow \ 12-3\sqrt{2} +4\sqrt{7} -\sqrt{14}}

Hence, the above expression is the solution.

Question 04
Solve the below expression.
\mathtt{\left(\sqrt{5} -2\right)\left(\sqrt{7} -\sqrt{6}\right)}

Solution
Simplifying the expression;

\mathtt{\Longrightarrow \left(\sqrt{5} -2\right)\left(\sqrt{7} -\sqrt{6}\right)}\\\ \\ \mathtt{\Longrightarrow \sqrt{5}\sqrt{7} -\sqrt{5}\sqrt{6} -2\sqrt{7} +2\sqrt{6}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{35} -\sqrt{30} -2\sqrt{7} +2\sqrt{6}}

Hence, the above expression is the solution.

Question 05
Simplify the below expression.
\mathtt{\left( -11+\sqrt{11}\right)\left( 2+\sqrt{2}\right)}

Solution
\mathtt{\Longrightarrow \left( -11+\sqrt{11}\right)\left( 2+\sqrt{2}\right)}\\\ \\ \mathtt{\Longrightarrow -11\times 2-11\sqrt{2} +2\sqrt{11} +\sqrt{11}\sqrt{2}}\\\ \\ \mathtt{\Longrightarrow -22-11\sqrt{2} +2\sqrt{11} +\sqrt{22}}

Next chapter : How to rationalize the denominator with square roots ?