Algebra Examples

$\frac{2 t^{3}}{\sqrt{4 t^{5}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $4 t^{5}$ as ${(2 t^{2})}^{2} t$ .

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Step 1.1.1

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

$\frac{2 t^{3}}{\sqrt{2^{2} t^{5}}}$

Step 1.1.2

Factor out $t^{4}$ .

$\frac{2 t^{3}}{\sqrt{2^{2} (t^{4} t)}}$

Step 1.1.3

Rewrite $t^{4}$ as ${(t^{2})}^{2}$ .

$\frac{2 t^{3}}{\sqrt{2^{2} ({(t^{2})}^{2} t)}}$

Step 1.1.4

Rewrite $2^{2} {(t^{2})}^{2}$ as ${(2 t^{2})}^{2}$ .

$\frac{2 t^{3}}{\sqrt{{(2 t^{2})}^{2} t}}$

Step 1.2

Pull terms out from under the radical.

$\frac{2 t^{3}}{2 t^{2} \sqrt{t}}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $2$ .

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Step 2.1.1

Cancel the common factor.

$\frac{2 t^{3}}{2 t^{2} \sqrt{t}}$

Step 2.1.2

Rewrite the expression.

$\frac{t^{3}}{t^{2} \sqrt{t}}$

Step 2.2

Cancel the common factor of $t^{3}$ and $t^{2}$ .

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Step 2.2.1

Factor $t^{2}$ out of $t^{3}$ .

$\frac{t^{2} t}{t^{2} \sqrt{t}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $t^{2}$ out of $t^{2} \sqrt{t}$ .

$\frac{t^{2} t}{t^{2} (\sqrt{t})}$

Step 2.2.2.2

Cancel the common factor.

$\frac{t^{2} t}{t^{2} \sqrt{t}}$

Step 2.2.2.3

Rewrite the expression.

$\frac{t}{\sqrt{t}}$

Step 3

Multiply $\frac{t}{\sqrt{t}}$ by $\frac{\sqrt{t}}{\sqrt{t}}$ .

$\frac{t}{\sqrt{t}} \cdot \frac{\sqrt{t}}{\sqrt{t}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{t}{\sqrt{t}}$ by $\frac{\sqrt{t}}{\sqrt{t}}$ .

$\frac{t \sqrt{t}}{\sqrt{t} \sqrt{t}}$

Step 4.2

Raise $\sqrt{t}$ to the power of $1$ .

$\frac{t \sqrt{t}}{{\sqrt{t}}^{1} \sqrt{t}}$

Step 4.3

Raise $\sqrt{t}$ to the power of $1$ .

$\frac{t \sqrt{t}}{{\sqrt{t}}^{1} {\sqrt{t}}^{1}}$

Step 4.4

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

$\frac{t \sqrt{t}}{{\sqrt{t}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{t \sqrt{t}}{{\sqrt{t}}^{2}}$

Step 4.6

Rewrite ${\sqrt{t}}^{2}$ as $t$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\frac{t \sqrt{t}}{{(t^{\frac{1}{2}})}^{2}}$

Step 4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{t \sqrt{t}}{t^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{t \sqrt{t}}{t^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{t \sqrt{t}}{t^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{t \sqrt{t}}{t^{1}}$

Step 4.6.5

Simplify.

$\frac{t \sqrt{t}}{t}$

Step 5

Cancel the common factor of $t$ .

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Step 5.1

Cancel the common factor.

$\frac{t \sqrt{t}}{t}$

Step 5.2

Divide $\sqrt{t}$ by $1$ .

$\sqrt{t}$