Pre-Algebra Examples

$3 \sqrt{12} \cdot \sqrt{27}$

Step 1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.1

Factor $4$ out of $12$ .

$3 \sqrt{4 (3)} \cdot \sqrt{27}$

Step 1.2

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

$3 \sqrt{2^{2} \cdot 3} \cdot \sqrt{27}$

Step 2

Pull terms out from under the radical.

$3 (2 \sqrt{3}) \cdot \sqrt{27}$

Step 3

Multiply $2$ by $3$ .

$6 \sqrt{3} \cdot \sqrt{27}$

Step 4

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

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Step 4.1

Factor $9$ out of $27$ .

$6 \sqrt{3} \cdot \sqrt{9 (3)}$

Step 4.2

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

$6 \sqrt{3} \cdot \sqrt{3^{2} \cdot 3}$

Step 5

Pull terms out from under the radical.

$6 \sqrt{3} \cdot (3 \sqrt{3})$

Step 6

Multiply $6 \sqrt{3} (3 \sqrt{3})$ .

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Step 6.1

Multiply $3$ by $6$ .

$18 \sqrt{3} \sqrt{3}$

Step 6.2

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

$18 ({\sqrt{3}}^{1} \sqrt{3})$

Step 6.3

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

$18 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 6.4

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

$18 {\sqrt{3}}^{1 + 1}$

Step 6.5

Add $1$ and $1$ .

$18 {\sqrt{3}}^{2}$

Step 7

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

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Step 7.1

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

$18 {(3^{\frac{1}{2}})}^{2}$

Step 7.2

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

$18 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 7.3

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

$18 \cdot 3^{\frac{2}{2}}$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

$18 \cdot 3^{\frac{2}{2}}$

Step 7.4.2

Rewrite the expression.

$18 \cdot 3^{1}$

Step 7.5

Evaluate the exponent.

$18 \cdot 3$

Step 8

Multiply $18$ by $3$ .

$54$