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 \(\displaystyle\sqrt{{{5}}}{\left\lbrace{x}\right\rbrace}+\sqrt{{{5}}}{\left\lbrace{y}\right\rbrace}+\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}\)

Answers (1)

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 given expression is already in simplified form as radicals are not like radicals.
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 \(\displaystyle\sqrt{{{5}}}{\left\lbrace{x}\right\rbrace}+\sqrt{{{5}}}{\left\lbrace{y}\right\rbrace}+\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}\)
0

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