Simplify f(x)=x square root of x^2* cube root of x^3* fourth root of x^5

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

Step 1

Factor out $x^{4}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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

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

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

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

$f (x) = x \sqrt{x^{2} \sqrt[3]{x^{3 + 1} \sqrt[4]{x}}}$

Add $3$ and $1$ .

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

Step 4

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

Factor out $x^{3}$ .

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

Add parentheses.

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

Step 5

Pull terms out from under the radical.

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

Step 6

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

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

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

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

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

$f (x) = x \sqrt{x^{2 + 1} \sqrt[3]{x \sqrt[4]{x}}}$

Add $2$ and $1$ .

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

Step 7

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

Factor out $x^{2}$ .

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

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

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

Add parentheses.

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

Step 8

Pull terms out from under the radical.

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

Step 9

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Simplify f(x)=x square root of x^2* cube root of x^3* fourth root of x^5

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