Algebra Examples

$\sqrt{10 x} = 3 \sqrt{x + 1}$

Step 1

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

${\sqrt{10 x}}^{2} = {(3 \sqrt{x + 1})}^{2}$

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 $\sqrt{10 x}$ as ${(10 x)}^{\frac{1}{2}}$ .

${({(10 x)}^{\frac{1}{2}})}^{2} = {(3 \sqrt{x + 1})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(10 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(10 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(10 x)}^{\frac{1}{2} \cdot 2} = {(3 \sqrt{x + 1})}^{2}$

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.

${(10 x)}^{\frac{1}{2} \cdot 2} = {(3 \sqrt{x + 1})}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(10 x)}^{1} = {(3 \sqrt{x + 1})}^{2}$

Step 2.2.1.2

Simplify.

$10 x = {(3 \sqrt{x + 1})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(3 \sqrt{x + 1})}^{2}$ .

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Step 2.3.1.1

Simplify the expression.

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Step 2.3.1.1.1

Apply the product rule to $3 \sqrt{x + 1}$ .

$10 x = 3^{2} {\sqrt{x + 1}}^{2}$

Step 2.3.1.1.2

Raise $3$ to the power of $2$ .

$10 x = 9 {\sqrt{x + 1}}^{2}$

Step 2.3.1.2

Rewrite ${\sqrt{x + 1}}^{2}$ as $x + 1$ .

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Step 2.3.1.2.1

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

$10 x = 9 {({(x + 1)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2.2

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

$10 x = 9 {(x + 1)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.2.3

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

$10 x = 9 {(x + 1)}^{\frac{2}{2}}$

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.

$10 x = 9 {(x + 1)}^{\frac{2}{2}}$

Step 2.3.1.2.4.2

Rewrite the expression.

$10 x = 9 {(x + 1)}^{1}$

Step 2.3.1.2.5

Simplify.

$10 x = 9 (x + 1)$

Step 2.3.1.3

Apply the distributive property.

$10 x = 9 x + 9 \cdot 1$

Step 2.3.1.4

Multiply $9$ by $1$ .

$10 x = 9 x + 9$

Step 3

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

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Step 3.1

Subtract $9 x$ from both sides of the equation.

$10 x - 9 x = 9$

Step 3.2

Subtract $9 x$ from $10 x$ .

$x = 9$