# To determine: To calculate: a) The indicated operation sqrt[4]{cd}cdotsqrt[5]{c^{2}} b) The indicated operation sqrt[4]{ysqrt[3]{y}} c) The indicated operation sqrt[5]{x}+sqrt[5]{y}+sqrt[4]{x}

Question
To determine:
To calculate:
a) The indicated operation $$\displaystyle\sqrt{{{4}}}{\left\lbrace{c}{d}\right\rbrace}\cdot\sqrt{{{5}}}{\left\lbrace{c}^{{{2}}}\right\rbrace}$$
b) The indicated operation $$\displaystyle\sqrt{{{4}}}{\left\lbrace{y}\sqrt{{{3}}}{\left\lbrace{y}\right\rbrace}\right\rbrace}$$
c) The indicated operation $$\displaystyle\sqrt{{{5}}}{\left\lbrace{x}\right\rbrace}+\sqrt{{{5}}}{\left\lbrace{y}\right\rbrace}+\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}$$

2021-02-12
Step 1 a) Formula Given: $$\displaystyle{a}^{{{m}}}\cdot\ {a}^{{{n}}}={a}^{{{m}+{n}}}$$
Consider the given expression $$\displaystyle\sqrt{{{4}}}{\left\lbrace{c}{d}\right\rbrace}\cdot\sqrt{{{5}}}{\left\lbrace{c}^{{{2}}}\right\rbrace}$$
Rewrite the expression with rational exponents
$$\displaystyle\sqrt{{{4}}}{\left\lbrace{c}{d}\right\rbrace}\cdot\sqrt{{{5}}}{\left\lbrace{c}^{{{2}}}\right\rbrace}$$
$$\displaystyle={\left({c}{d}\cdot{\left({c}{d}\right)}\right)}$$
$$\displaystyle={c}^{{\frac{{1}}{{4}}}}\cdot\ {d}^{{\frac{{1}}{{4}}}}\cdot\ {c}^{{\frac{{2}}{{5}}}}$$
Combine the like bases and use the property $$\displaystyle{a}^{{{m}}}\cdot\ {a}^{{{n}}}={a}^{{{m}+{n}}}$$
$$\displaystyle{c}^{{\frac{{1}}{{4}}}}\cdot\ {d}^{{\frac{{1}}{{4}}}}\cdot\ {c}^{{\frac{{2}}{{5}}}}$$
$$\displaystyle={c}^{{\frac{{1}}{{4}}+\frac{{2}}{{5}}}}\cdot\ {d}^{{\frac{{1}}{{4}}}}$$
$$\displaystyle=\sqrt{{{20}}}{\left\lbrace{c}^{{{13}}}{d}^{{{5}}}\right\rbrace}$$
Therefore, the value of $$\displaystyle\sqrt{{{4}}}{\left\lbrace{c}{d}\right\rbrace}\cdot\sqrt{{{5}}}{\left\lbrace{c}^{{{2}}}\right\rbrace}$$ is $$\displaystyle\sqrt{{{20}}}{\left\lbrace{c}^{{{13}}}{d}^{{{5}}}\right\rbrace}$$
Step 2
b) Formula Given: $$\displaystyle{\left({a}^{{{m}}}\right)}^{{{n}}}={a}^{{{m}{n}}}$$
$$\displaystyle{a}^{{{m}}}\cdot\ {a}^{{{n}}}={a}^{{{m}+{n}}}$$
Consider the given expression $$\displaystyle\sqrt{{{4}}}{\left\lbrace{y}\sqrt{{{3}}}{\left\lbrace{y}\right\rbrace}\right\rbrace}$$
Rewrite the expression with rational exponents
$$\displaystyle\sqrt{{{4}}}{\left\lbrace{y}\sqrt{{{3}}}{\left\lbrace{y}\right\rbrace}\right\rbrace}$$
$$\displaystyle={\left(_{\left\lbrace.\right\rbrace}^{{{y}}}\right.}$$
Use the property $$\displaystyle{a}^{{{m}}}\cdot\ {a}^{{{n}}}={a}^{{{m}+{n}}}$$ to get,
$$\displaystyle={\left({y}^{{\frac{{4}}{{3}}}}\right)}^{{\frac{{1}}{{4}}}}$$
Use the property $$\displaystyle{\left({a}^{{{m}}}\right)}^{{{n}}}={a}^{{{m}{n}}}$$ to get,
$$\displaystyle={y}^{{\frac{{1}}{{3}}}}$$
$$\displaystyle=\sqrt{{{3}}}{\left\lbrace{y}\right\rbrace}$$
Therefore, the value of $$\displaystyle\sqrt{{{4}}}{\left\lbrace{y}\sqrt{{{3}}}{\left\lbrace{y}\right\rbrace}\right\rbrace}$$ is $$\displaystyle\sqrt{{{3}}}{\left\lbrace{y}\right\rbrace}$$
Step 3 NKS c) Consider the given expression $$\displaystyle\sqrt{{{5}}}{\left\lbrace{x}\right\rbrace}+\sqrt{{{5}}}{\left\lbrace{y}\right\rbrace}+\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}$$
The first two terms have different indices and the first and third terms have different radicands.
Therefore, the value of $$\displaystyle\sqrt{{{5}}}{\left\lbrace{x}\right\rbrace}+\sqrt{{{5}}}{\left\lbrace{y}\right\rbrace}+\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}$$ is $$\displaystyle\sqrt{{{5}}}{\left\lbrace{x}\right\rbrace}+\sqrt{{{5}}}{\left\lbrace{y}\right\rbrace}+\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}$$

### Relevant Questions

To determine:
The radicand of the expression is $$\displaystyle\sqrt{{{3}}}\cdot\sqrt{{{3}}}{\left\lbrace{2}\right\rbrace}$$
To simplify:
The given expression $$\displaystyle\sqrt{{{3}}}{\left\lbrace{\frac{{{54}{x}^{{{2}}}{y}^{{{4}}}}}{{{2}{x}^{{{5}}}{y}}}}\right\rbrace}$$
Perform the indicated operation and express all answers with a rational denominator $$2a^{2}b\sqrt[3]{4a^{3}b}\ \cdot\ -6ab^{2}\sqrt[3]{18ab^{4}}$$
To determine:
The simplified value of the radical expression $$\displaystyle\sqrt{{{3}}}{\left\lbrace{2}{a}\right\rbrace}\cdot\sqrt{{{2}{a}}}$$
To calculate:
To convert the radical expression $$\displaystyle\sqrt{{{3}}}{\left\lbrace{x}^{{{6}}}\ {y}^{{{9}}}\right\rbrace}$$ to rational exponent form and simlify.
We need to calculate: The simplified form of $$\sqrt[5]{x^{2}y^{2}}\ \cdot\ \sqrt[4]{x}$$
To calculate the radical expression $$\displaystyle\sqrt{{{3}}}{\left\lbrace{x}^{{{12}}}\right\rbrace}$$ to rational exponent form and simplify.
Radical simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers.
a) $$\displaystyle\sqrt{{{6}}}{\left\lbrace{y}^{{{5}}}\right\rbrace}\sqrt{{{3}}}{\left\lbrace{y}^{{{2}}}\right\rbrace}$$
b) $$\displaystyle{\left({5}\sqrt{{{3}}}{\left\lbrace{x}\right\rbrace}\right)}{\left({2}\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}\right)}$$
EXAMPLE: X $$-1/4$$. If X = 4, this gives _0.7071__. Compute to two decimal places or more.
a. X $$-1/5$$ , When X = 2, this gives ____.
c. X $$1/2$$, When X = 64, this gives ____.
d. X $$3/4$$ , When X = 2, this gives ____.
Simplify the expression, answering with rational exponents and not radicals. To enter $$\displaystyle{x}^{{\frac{{m}}{{n}}}},$$ type $$\displaystyle{x}^{{{\left(\frac{{m}}{{n}}\right)}}}.$$
$$\displaystyle\sqrt{{{3}}}{\left\lbrace{27}{m}^{{{2}}}\right\rbrace}=?$$