Calculus Examples

$\sqrt{27} a^{2} \sqrt{75}$

Step 1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.1

Factor $9$ out of $27$ .

$\sqrt{9 (3)} a^{2} \sqrt{75}$

Step 1.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} \cdot 3} a^{2} \sqrt{75}$

Step 2

Pull terms out from under the radical.

$3 \sqrt{3} a^{2} \sqrt{75}$

Step 3

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 3.1

Factor $25$ out of $75$ .

$3 \sqrt{3} a^{2} \sqrt{25 (3)}$

Step 3.2

Rewrite $25$ as $5^{2}$ .

$3 \sqrt{3} a^{2} \sqrt{5^{2} \cdot 3}$

Step 4

Pull terms out from under the radical.

$3 \sqrt{3} a^{2} (5 \sqrt{3})$

Step 5

Multiply $3 \sqrt{3} a^{2} (5 \sqrt{3})$ .

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Step 5.1

Multiply $5$ by $3$ .

$15 \sqrt{3} a^{2} \sqrt{3}$

Step 5.2

Raise $\sqrt{3}$ to the power of $1$ .

$15 a^{2} ({\sqrt{3}}^{1} \sqrt{3})$

Step 5.3

Raise $\sqrt{3}$ to the power of $1$ .

$15 a^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 5.4

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

$15 a^{2} {\sqrt{3}}^{1 + 1}$

Step 5.5

Add $1$ and $1$ .

$15 a^{2} {\sqrt{3}}^{2}$

Step 6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$15 a^{2} {(3^{\frac{1}{2}})}^{2}$

Step 6.2

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

$15 a^{2} \cdot 3^{\frac{1}{2} \cdot 2}$

Step 6.3

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

$15 a^{2} \cdot 3^{\frac{2}{2}}$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

$15 a^{2} \cdot 3^{\frac{2}{2}}$

Step 6.4.2

Rewrite the expression.

$15 a^{2} \cdot 3^{1}$

Step 6.5

Evaluate the exponent.

$15 a^{2} \cdot 3$

Step 7

Multiply $3$ by $15$ .

$45 a^{2}$