necessaryh

2021-01-02

We need to calculate: The simplified form of $\sqrt[5]{{x}^{2}{y}^{2}}\text{}\cdot \text{}\sqrt[4]{x}$

rogreenhoxa8

Skilled2021-01-03Added 109 answers

Given:

The expression is$\sqrt[5]{{x}^{2}{y}^{2}}\text{}\cdot \text{}\sqrt[4]{x}$

Formula used:

For a real number a when m is even,

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

Calculation:

Consider the provided expression:

$\sqrt[5]{{x}^{2}{y}^{2}}\text{}\cdot \text{}\sqrt[4]{x}$

Since, the radicals have different indices, the product property of radicals cannot be applied directly.

So, rewrite the expression with rational exponents:

$\sqrt[5]{{x}^{2}{y}^{2}}\text{}\cdot \text{}\sqrt[4]{x}=({x}^{3}y2{)}^{\frac{1}{5}}\text{}\cdot \text{}{x}^{\frac{1}{4}}$

Apply the power rule of exponents:

$({x}^{3}{y}^{2}{)}^{\frac{1}{5}}\text{}\cdot \text{}x\left(\frac{1}{4}\right)={x}^{\frac{3}{5}}\text{}{y}^{\frac{2}{5}}\text{}{x}^{\frac{1}{4}}$

Add exponents of like bases:

${x}^{\frac{3}{5}}\text{}{y}^{\frac{2}{5}}\text{}{x}^{\frac{1}{4}}=x(\frac{3}{5}\text{}+\text{}\frac{1}{4}){y}^{\frac{2}{5}}$

$=x(\frac{12}{20}\text{}+\text{}\frac{5}{20}){y}^{\frac{2}{5}}$

$={x}^{\frac{17}{20}}{y}^{\frac{2}{5}}$

Writing each exponent with the same denominator, so that in radical notation, the factors will have the same index.

${x}^{\frac{17}{20}}{y}^{\frac{2}{5}}={x}^{\frac{17}{20}}{y}^{\frac{8}{20}}$

$=\text{}\frac{({x}^{17}{y}^{8}{)}^{1}}{20}$

Convert to radical notation:

$\frac{({x}^{17}{y}^{8}{)}^{1}}{20}=\sqrt[20]{{x}^{17}{y}^{8}}$

The expression is

Formula used:

For a real number a when m is even,

Calculation:

Consider the provided expression:

Since, the radicals have different indices, the product property of radicals cannot be applied directly.

So, rewrite the expression with rational exponents:

Apply the power rule of exponents:

Add exponents of like bases:

Writing each exponent with the same denominator, so that in radical notation, the factors will have the same index.

Convert to radical notation:

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

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