Algebra Examples

$f (x) = x \sqrt{x^{2} \cdot \sqrt[3]{x^{3} \cdot \sqrt[4]{x^{5}}}}$

Step 1

Write $f (x) = x \sqrt{x^{2} \cdot \sqrt[3]{x^{3} \cdot \sqrt[4]{x^{5}}}}$ as an equation.

$y = x \sqrt{x^{2} \cdot \sqrt[3]{x^{3} \cdot \sqrt[4]{x^{5}}}}$

Step 2

Simplify $x \sqrt{x^{2} \cdot \sqrt[3]{x^{3} \cdot \sqrt[4]{x^{5}}}}$ .

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Step 2.1

Factor out $x^{4}$ .

$y = x \sqrt{x^{2} \sqrt[3]{x^{3} \sqrt[4]{x^{4} x}}}$

Step 2.2

Pull terms out from under the radical.

$y = x \sqrt{x^{2} \sqrt[3]{x^{3} x \sqrt[4]{x}}}$

Step 2.3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 2.3.1

Multiply $x^{3}$ by $x$ .

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Step 2.3.1.1

Raise $x$ to the power of $1$ .

$y = x \sqrt{x^{2} \sqrt[3]{x^{3} x^{1} \sqrt[4]{x}}}$

Step 2.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x \sqrt{x^{2} \sqrt[3]{x^{3 + 1} \sqrt[4]{x}}}$

Step 2.3.2

Add $3$ and $1$ .

$y = x \sqrt{x^{2} \sqrt[3]{x^{4} \sqrt[4]{x}}}$

Step 2.4

Rewrite $x^{4} \sqrt[4]{x}$ as $x^{3} (x \sqrt[4]{x})$ .

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Step 2.4.1

Factor out $x^{3}$ .

$y = x \sqrt{x^{2} \sqrt[3]{x^{3} x \sqrt[4]{x}}}$

Step 2.4.2

Add parentheses.

$y = x \sqrt{x^{2} \sqrt[3]{x^{3} (x \sqrt[4]{x})}}$

Step 2.5

Pull terms out from under the radical.

$y = x \sqrt{x^{2} x \sqrt[3]{x \sqrt[4]{x}}}$

Step 2.6

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.6.1

Multiply $x^{2}$ by $x$ .

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Step 2.6.1.1

Raise $x$ to the power of $1$ .

$y = x \sqrt{x^{2} x^{1} \sqrt[3]{x \sqrt[4]{x}}}$

Step 2.6.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x \sqrt{x^{2 + 1} \sqrt[3]{x \sqrt[4]{x}}}$

Step 2.6.2

Add $2$ and $1$ .

$y = x \sqrt{x^{3} \sqrt[3]{x \sqrt[4]{x}}}$

Step 2.7

Rewrite $x^{3} \sqrt[3]{x \sqrt[4]{x}}$ as $x^{2} (x \sqrt[3]{x \sqrt[2^{2}]{x}})$ .

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Step 2.7.1

Factor out $x^{2}$ .

$y = x \sqrt{x^{2} x \sqrt[3]{x \sqrt[4]{x}}}$

Step 2.7.2

Rewrite $4$ as $2^{2}$ .

$y = x \sqrt{x^{2} x \sqrt[3]{x \sqrt[2^{2}]{x}}}$

Step 2.7.3

Add parentheses.

$y = x \sqrt{x^{2} (x \sqrt[3]{x \sqrt[2^{2}]{x}})}$

Step 2.8

Pull terms out from under the radical.

$y = x (x \sqrt{x \sqrt[3]{x \sqrt[2^{2}]{x}}})$

Step 2.9

Multiply $x$ by $x$ by adding the exponents.

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Step 2.9.1

Move $x$ .

$y = x \cdot x \sqrt{x \sqrt[3]{x \sqrt[2^{2}]{x}}}$

Step 2.9.2

Multiply $x$ by $x$ .

$y = x^{2} \sqrt{x \sqrt[3]{x \sqrt[2^{2}]{x}}}$