# The simplified form of the expression displaystyle{sqrt[{{4}}]{{{c}{d}^{2}}}}times{sqrt[{{3}}]{{{c}^{2}{d}}}}.

The simplified form of the expression
$\sqrt[4]{c{d}^{2}}×\sqrt[3]{{c}^{2}d}.$
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Formula used:
If a and b are real numbers and is an integer, the product property is true provided that the radicals are real numbers.
$\sqrt[n]{a}×\sqrt[n]{b}=\sqrt[n]{ab}$
If m and n are integers and is an integer, then
$\sqrt[n]{{a}^{m}}={a}^{mn}$
Calculation:
Consider the expression,
$\sqrt[4]{c{d}^{2}}×\sqrt[3]{{c}^{2}d}$
Rewrite the provided expression as rational exponents.
$\sqrt[4]{c{d}^{2}}×\sqrt[3]{{c}^{2}d}={\left(c{d}^{2}\right)}^{1\text{/}4}×{\left({c}^{2}d\right)}^{1\text{/}3}$
Use the product property.
${\left(c{d}^{2}\right)}^{1\text{/}4}×{\left({c}^{2}d\right)}^{1\text{/}3}={\left(c\right)}^{1\text{/}4}×{\left({d}^{2}\right)}^{1\text{/}4}×{\left({c}^{2}\right)}^{1\text{/}3}×{\left(d\right)}^{1\text{/}3}$
Use the formula $\sqrt[n]{{a}^{m}}={a}^{m\text{/}n}$ and simplify the expression.
${\left(c\right)}^{1\text{/}4}×{\left({d}^{2}\right)}^{1\text{/}4}×{\left({c}^{2}\right)}^{1\text{/}3}×{\left(d\right)}^{1\text{/}3}={c}^{1\text{/}4}×{d}^{2\text{/}4}×{c}^{2\text{/}3}×{d}^{1\text{/}3}$
Add the powers of the same bases.
${c}^{1\text{/}4}×{d}^{2\text{/}4}×{c}^{2\text{/}3}×{d}^{1\text{/}3}=c\left(\frac{1}{4}\right)+\left(\frac{2}{4}\right)×d\left(\frac{2}{4}\right)+\left(\frac{1}{3}\right)$
$={c}^{\frac{11}{12}}×{d}^{\frac{10}{12}}$
The obtained expression with rational exponents can be rewritten into radicals as,
$={c}^{\frac{11}{12}}×{d}^{\frac{10}{12}}={\left({c}^{11}\right)}^{\frac{1}{12}}×{\left({d}^{10}\right)}^{\frac{1}{12}}$
$=\sqrt[12]{{c}^{11}{d}^{10}}$
Answer: $\sqrt[4]{c{d}^{2}}×\sqrt[3]{{c}^{2}d}is\sqrt[12]{{c}^{11}{d}^{10}}$
Jeffrey Jordon