Algebra Examples

$\sqrt{9 k - 9} = \sqrt{6 k + 18}$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{9 k - 9}}^{2} = {\sqrt{6 k + 18}}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 k - 9}$ as ${(9 k - 9)}^{\frac{1}{2}}$ .

${({(9 k - 9)}^{\frac{1}{2}})}^{2} = {\sqrt{6 k + 18}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(9 k - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(9 k - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(9 k - 9)}^{\frac{1}{2} \cdot 2} = {\sqrt{6 k + 18}}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(9 k - 9)}^{\frac{1}{2} \cdot 2} = {\sqrt{6 k + 18}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(9 k - 9)}^{1} = {\sqrt{6 k + 18}}^{2}$

Step 2.2.1.2

Simplify.

$9 k - 9 = {\sqrt{6 k + 18}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{6 k + 18}}^{2}$ .

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Step 2.3.1.1

Factor $6$ out of $6 k + 18$ .

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Step 2.3.1.1.1

Factor $6$ out of $6 k$ .

$9 k - 9 = {\sqrt{6 (k) + 18}}^{2}$

Step 2.3.1.1.2

Factor $6$ out of $18$ .

$9 k - 9 = {\sqrt{6 k + 6 \cdot 3}}^{2}$

Step 2.3.1.1.3

Factor $6$ out of $6 k + 6 \cdot 3$ .

$9 k - 9 = {\sqrt{6 (k + 3)}}^{2}$

Step 2.3.1.2

Rewrite ${\sqrt{6 (k + 3)}}^{2}$ as $6 (k + 3)$ .

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Step 2.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 (k + 3)}$ as ${(6 (k + 3))}^{\frac{1}{2}}$ .

$9 k - 9 = {({(6 (k + 3))}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2.2

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

$9 k - 9 = {(6 (k + 3))}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.2.3

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

$9 k - 9 = {(6 (k + 3))}^{\frac{2}{2}}$

Step 2.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.2.4.1

Cancel the common factor.

$9 k - 9 = {(6 (k + 3))}^{\frac{2}{2}}$

Step 2.3.1.2.4.2

Rewrite the expression.

$9 k - 9 = {(6 (k + 3))}^{1}$

Step 2.3.1.2.5

Simplify.

$9 k - 9 = 6 (k + 3)$

Step 2.3.1.3

Apply the distributive property.

$9 k - 9 = 6 k + 6 \cdot 3$

Step 2.3.1.4

Multiply $6$ by $3$ .

$9 k - 9 = 6 k + 18$

Step 3

Solve for $k$ .

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Step 3.1

Move all terms containing $k$ to the left side of the equation.

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Step 3.1.1

Subtract $6 k$ from both sides of the equation.

$9 k - 9 - 6 k = 18$

Step 3.1.2

Subtract $6 k$ from $9 k$ .

$3 k - 9 = 18$

Step 3.2

Move all terms not containing $k$ to the right side of the equation.

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Step 3.2.1

Add $9$ to both sides of the equation.

$3 k = 18 + 9$

Step 3.2.2

Add $18$ and $9$ .

$3 k = 27$

Step 3.3

Divide each term in $3 k = 27$ by $3$ and simplify.

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Step 3.3.1

Divide each term in $3 k = 27$ by $3$ .

$\frac{3 k}{3} = \frac{27}{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{3 k}{3} = \frac{27}{3}$

Step 3.3.2.1.2

Divide $k$ by $1$ .

$k = \frac{27}{3}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $27$ by $3$ .

$k = 9$