# Radicals and Exponents Evaluate each expression: a) frac{sqrt{132}}{sqrt{3}} b) sqrt[3]{2}sqrt[3]{32} c) sqrt[4]{frac{1}{4}}sqrt[4]{frac{1}{64}}

Question
Radicals and Exponents Evaluate each expression:
a) $$\displaystyle{\frac{{\sqrt{{{132}}}}}{{\sqrt{{{3}}}}}}$$
b) $$\displaystyle\sqrt{{{3}}}{\left\lbrace{2}\right\rbrace}\sqrt{{{3}}}{\left\lbrace{32}\right\rbrace}$$
c) $$\displaystyle\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{4}}}}\right\rbrace}\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{64}}}}\right\rbrace}$$

2021-01-09
a) Formula used:
Power of nth roots:
$$\displaystyle\sqrt{{{n}}}{\left\lbrace{\frac{{{a}}}{{{b}}}}\right\rbrace}={\frac{{\sqrt{{{n}}}{\left\lbrace{a}\right\rbrace}}}{{\sqrt{{{n}}}{\left\lbrace{a}\right\rbrace}}}}$$
Where n is any positive integer and a and b are bases.
Calculation:
The given exponential expression is $$\displaystyle{\frac{{\sqrt{{{132}}}}}{{\sqrt{{{3}}}}}}$$
Use the above-mentioned formula and calculate the value of $$\displaystyle{\frac{{\sqrt{{{132}}}}}{{\sqrt{{{3}}}}}}$$ as shown below.
$$\displaystyle{\frac{{\sqrt{{{132}}}}}{{\sqrt{{{3}}}}}}={\frac{{\sqrt{{{11}\cdot{2}\cdot{2}\cdot{3}}}}}{{\sqrt{{{3}}}}}}$$
$$\displaystyle={\frac{{{2}\sqrt{{{11}}}\cdot\sqrt{{{3}}}}}{{\sqrt{{{3}}}}}}$$
$$\displaystyle={2}\sqrt{{{11}}}$$
Thus, the value of exponential is $$\displaystyle{2}\sqrt{{{11}}}$$
b) Calculation:
Use the above mentioned formula and simplify the given expression as shown below.
$$\displaystyle\sqrt{{{3}}}{\left\lbrace{2}\right\rbrace}\sqrt{{{3}}}{\left\lbrace{32}\right\rbrace}=\sqrt{{{3}}}{\left\lbrace{2}\right\rbrace}\sqrt{{{3}}}{\left\lbrace{2}\cdot{2}\cdot{2}\cdot{2}\cdot{2}\right\rbrace}$$
$$\displaystyle={\left({2}^{{{6}}}\right)}^{{{\frac{{{1}}}{{{3}}}}}}$$
$$\displaystyle={2}^{{{2}}}$$
$$\displaystyle={4}$$
Thus, the value of exponential expression is 4.
c) Calculation:
$$\displaystyle\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{4}}}}\right\rbrace}\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{64}}}}\right\rbrace}=\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{4}}}}\right\rbrace}\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{4}\cdot{4}\cdot{4}}}}\right\rbrace}$$
$$\displaystyle=\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{\left({4}\right)}^{{{4}}}}}}\right\rbrace}$$
$$\displaystyle={\frac{{{1}}}{{{\left({4}^{{{4}}}\right)}^{{\frac{{{1}}}{{{4}}}}}}}}$$
$$\displaystyle={\frac{{{1}}}{{{4}}}}$$
The value of exponential is $$\displaystyle{\frac{{{1}}}{{{4}}}}.$$

### Relevant Questions

Radicals and Exponents Evaluate each expression:
a) $$\displaystyle{2}\sqrt{{{3}}}{\left\lbrace{81}\right\rbrace}$$
b) $$\displaystyle{\frac{{\sqrt{{{18}}}}}{{\sqrt{{{25}}}}}}$$
c) $$\displaystyle\sqrt{{{\frac{{{12}}}{{{49}}}}}}$$
Use the properties of logarithms to rewrite each expression as the logarithm of a single expression. Be sure to use positive exponents and avoid radicals.
a. $$2\ln4x^{3}\ +\ 3\ln\ y\ -\ \frac{1}{3}\ln\ z^{6}$$
b. $$\ln(x^{2}\ -\ 16)\ -\ \ln(x\ +\ 4)$$
Write each radical expression using exponents and each exponential expression using radicals.
$$\frac{Radical\ expression\ Exponential\ expression}{\sqrt[5]{5^{3}}}$$
Radicals and Exponents Simplify the expression.
$$\displaystyle{\frac{{{x}^{{{4}}}{\left({3}{x}\right)}^{{{2}}}}}{{{x}^{{{3}}}}}}$$
Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
$$\displaystyle{\frac{{\sqrt{{{4}}}{\left\lbrace{x}^{{{7}}}\right\rbrace}}}{{\sqrt{{{4}}}{\left\lbrace{x}^{{{3}}}\right\rbrace}}}}$$
Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
$$\displaystyle{\frac{{\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}$$
Rewrite each expression so that each term is in the form $$kx^{n},$$ where k is a real number, x is a positive real number, and n is a rational number.
a)$$x^{-\frac{2}{3}}\ \times\ x^{\frac{1}{3}}=?$$
b) $$\frac{10x^{\frac{1}{3}}}{2x^{2}}=?$$
c) $$(3x^{\frac{1}{4}})^{-2}=?$$
a) $$\displaystyle\sqrt{{{6}}}{\left\lbrace{y}^{{{5}}}\right\rbrace}\sqrt{{{3}}}{\left\lbrace{y}^{{{2}}}\right\rbrace}$$
b) $$\displaystyle{\left({5}\sqrt{{{3}}}{\left\lbrace{x}\right\rbrace}\right)}{\left({2}\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}\right)}$$
$$\displaystyle{5}{x}^{{{\frac{{{5}}}{{{2}}}}}}+{2}{x}^{{{\frac{{{1}}}{{{2}}}}}}+{x}^{{{\frac{{{3}}}{{{2}}}}}}$$
(a) $$\displaystyle{27}^{{{1}\text{/}{3}}}$$
(b) $$\displaystyle{\left(-{8}\right)}^{{{1}\text{/}{3}}}$$
(c) $$\displaystyle-{\left(\frac{1}{{8}}\right)}^{{{1}\text{/}{3}}}$$