Khaleesi Herbert

2021-01-05

The simplified form of the expression

$\sqrt[4]{c{d}^{2}}\times \sqrt[3]{{c}^{2}d}.$

grbavit

Skilled2021-01-06Added 109 answers

Formula used:

Product property of radicals:

If a and b are real numbers and$n\text{}\text{}1$ is an integer, the product property is true provided that the radicals are real numbers.

$\sqrt[n]{a}\times \sqrt[n]{b}=\sqrt[n]{ab}$

If m and n are integers and$n\text{}\text{}1$ is an integer, then

$\sqrt[n]{{a}^{m}}={a}^{mn}$

Calculation:

Consider the expression,

$\sqrt[4]{c{d}^{2}}\times \sqrt[3]{{c}^{2}d}$

Rewrite the provided expression as rational exponents.

$\sqrt[4]{c{d}^{2}}\times \sqrt[3]{{c}^{2}d}={\left(c{d}^{2}\right)}^{1\text{/}4}\times {\left({c}^{2}d\right)}^{1\text{/}3}$

Use the product property.

$\left(c{d}^{2}\right)}^{1\text{/}4}\times {\left({c}^{2}d\right)}^{1\text{/}3}={\left(c\right)}^{1\text{/}4}\times {\left({d}^{2}\right)}^{1\text{/}4}\times {\left({c}^{2}\right)}^{1\text{/}3}\times {\left(d\right)}^{1\text{/}3$

Use the formula$\sqrt[n]{{a}^{m}}={a}^{m\text{/}n}$ and simplify the expression.

$\left(c\right)}^{1\text{/}4}\times {\left({d}^{2}\right)}^{1\text{/}4}\times {\left({c}^{2}\right)}^{1\text{/}3}\times {\left(d\right)}^{1\text{/}3}={c}^{1\text{/}4}\times {d}^{2\text{/}4}\times {c}^{2\text{/}3}\times {d}^{1\text{/}3$

Add the powers of the same bases.

${c}^{1\text{/}4}\times {d}^{2\text{/}4}\times {c}^{2\text{/}3}\times {d}^{1\text{/}3}=c\left(\frac{1}{4}\right)+\left(\frac{2}{4}\right)\times d\left(\frac{2}{4}\right)+\left(\frac{1}{3}\right)$

$={c}^{\frac{11}{12}}\times {d}^{\frac{10}{12}}$

The obtained expression with rational exponents can be rewritten into radicals as,

$={c}^{\frac{11}{12}}\times {d}^{\frac{10}{12}}={\left({c}^{11}\right)}^{\frac{1}{12}}\times {\left({d}^{10}\right)}^{\frac{1}{12}}$

$=\sqrt[12]{{c}^{11}{d}^{10}}$

Answer:$\sqrt[4]{c{d}^{2}}\times \sqrt[3]{{c}^{2}d}is\sqrt[12]{{c}^{11}{d}^{10}}$

Product property of radicals:

If a and b are real numbers and

If m and n are integers and

Calculation:

Consider the expression,

Rewrite the provided expression as rational exponents.

Use the product property.

Use the formula

Add the powers of the same bases.

The obtained expression with rational exponents can be rewritten into radicals as,

Answer:

Jeffrey Jordon

Expert2021-10-26Added 2605 answers

Answer is given below (on video)

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