Algebra Examples

$\sqrt{3 k} (\sqrt{12 k} - 2 \sqrt{15 k^{2}})$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $12$ .

$\sqrt{3 k} (\sqrt{4 (3) k} - 2 \sqrt{15 k^{2}})$

Step 1.1.2

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

$\sqrt{3 k} (\sqrt{2^{2} \cdot 3 k} - 2 \sqrt{15 k^{2}})$

Step 1.1.3

Add parentheses.

$\sqrt{3 k} (\sqrt{2^{2} \cdot (3 k)} - 2 \sqrt{15 k^{2}})$

Step 1.2

Pull terms out from under the radical.

$\sqrt{3 k} (2 \sqrt{3 k} - 2 \sqrt{15 k^{2}})$

Step 1.3

Reorder $15$ and $k^{2}$ .

$\sqrt{3 k} (2 \sqrt{3 k} - 2 \sqrt{k^{2} \cdot 15})$

Step 1.4

Pull terms out from under the radical.

$\sqrt{3 k} (2 \sqrt{3 k} - 2 k \sqrt{15})$

Step 2

Simplify by multiplying through.

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Step 2.1

Apply the distributive property.

$\sqrt{3 k} (2 \sqrt{3 k}) + \sqrt{3 k} (- 2 k \sqrt{15})$

Step 2.2

Reorder.

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Step 2.2.1

Rewrite using the commutative property of multiplication.

$2 \sqrt{3 k} \sqrt{3 k} + \sqrt{3 k} (- 2 k \sqrt{15})$

Step 2.2.2

Rewrite using the commutative property of multiplication.

$2 \sqrt{3 k} \sqrt{3 k} - 2 \sqrt{15} \sqrt{3 k} k$

Step 3

Simplify each term.

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Step 3.1

Multiply $2 \sqrt{3 k} \sqrt{3 k}$ .

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

$2 {\sqrt{3 k}}^{1 + 1} - 2 \sqrt{15} \sqrt{3 k} k$

Step 3.1.4

Add $1$ and $1$ .

$2 {\sqrt{3 k}}^{2} - 2 \sqrt{15} \sqrt{3 k} k$

Step 3.2

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

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Step 3.2.1

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

$2 {({(3 k)}^{\frac{1}{2}})}^{2} - 2 \sqrt{15} \sqrt{3 k} k$

Step 3.2.2

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

$2 {(3 k)}^{\frac{1}{2} \cdot 2} - 2 \sqrt{15} \sqrt{3 k} k$

Step 3.2.3

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

$2 {(3 k)}^{\frac{2}{2}} - 2 \sqrt{15} \sqrt{3 k} k$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

$2 {(3 k)}^{\frac{2}{2}} - 2 \sqrt{15} \sqrt{3 k} k$

Step 3.2.4.2

Rewrite the expression.

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

Step 3.2.5

Simplify.

$2 (3 k) - 2 \sqrt{15} \sqrt{3 k} k$

Step 3.3

Multiply $3$ by $2$ .

$6 k - 2 \sqrt{15} \sqrt{3 k} k$

Step 3.4

Multiply $- 2 \sqrt{15} \sqrt{3 k}$ .

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Step 3.4.1

Combine using the product rule for radicals.

$6 k - 2 \sqrt{3 k \cdot 15} k$

Step 3.4.2

Multiply $15$ by $3$ .

$6 k - 2 \sqrt{45 k} k$

Step 3.5

Rewrite $45 k$ as $3^{2} \cdot (5 k)$ .

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Step 3.5.1

Factor $9$ out of $45$ .

$6 k - 2 \sqrt{9 (5) k} k$

Step 3.5.2

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

$6 k - 2 \sqrt{3^{2} \cdot 5 k} k$

Step 3.5.3

Add parentheses.

$6 k - 2 \sqrt{3^{2} \cdot (5 k)} k$

Step 3.6

Pull terms out from under the radical.

$6 k - 2 (3 \sqrt{5 k}) k$

Step 3.7

Multiply $3$ by $- 2$ .

$6 k - 6 \sqrt{5 k} k$