Basic Math Examples

$\sqrt{b^{2} - 13} = b - 1$

Step 1

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

${\sqrt{b^{2} - 13}}^{2} = {(b - 1)}^{2}$

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 $\sqrt{b^{2} - 13}$ as ${(b^{2} - 13)}^{\frac{1}{2}}$ .

${({(b^{2} - 13)}^{\frac{1}{2}})}^{2} = {(b - 1)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(b^{2} - 13)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(b^{2} - 13)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(b^{2} - 13)}^{\frac{1}{2} \cdot 2} = {(b - 1)}^{2}$

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.

${(b^{2} - 13)}^{\frac{1}{2} \cdot 2} = {(b - 1)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(b^{2} - 13)}^{1} = {(b - 1)}^{2}$

Step 2.2.1.2

Simplify.

$b^{2} - 13 = {(b - 1)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(b - 1)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(b - 1)}^{2}$ as $(b - 1) (b - 1)$ .

$b^{2} - 13 = (b - 1) (b - 1)$

Step 2.3.1.2

Expand $(b - 1) (b - 1)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$b^{2} - 13 = b (b - 1) - 1 (b - 1)$

Step 2.3.1.2.2

Apply the distributive property.

$b^{2} - 13 = b \cdot b + b \cdot - 1 - 1 (b - 1)$

Step 2.3.1.2.3

Apply the distributive property.

$b^{2} - 13 = b \cdot b + b \cdot - 1 - 1 b - 1 \cdot - 1$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $b$ by $b$ .

$b^{2} - 13 = b^{2} + b \cdot - 1 - 1 b - 1 \cdot - 1$

Step 2.3.1.3.1.2

Move $- 1$ to the left of $b$ .

$b^{2} - 13 = b^{2} - 1 \cdot b - 1 b - 1 \cdot - 1$

Step 2.3.1.3.1.3

Rewrite $- 1 b$ as $- b$ .

$b^{2} - 13 = b^{2} - b - 1 b - 1 \cdot - 1$

Step 2.3.1.3.1.4

Rewrite $- 1 b$ as $- b$ .

$b^{2} - 13 = b^{2} - b - b - 1 \cdot - 1$

Step 2.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

$b^{2} - 13 = b^{2} - b - b + 1$

Step 2.3.1.3.2

Subtract $b$ from $- b$ .

$b^{2} - 13 = b^{2} - 2 b + 1$

Step 3

Solve for $b$ .

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Step 3.1

Since $b$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$b^{2} - 2 b + 1 = b^{2} - 13$

Step 3.2

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

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Step 3.2.1

Subtract $b^{2}$ from both sides of the equation.

$b^{2} - 2 b + 1 - b^{2} = - 13$

Step 3.2.2

Combine the opposite terms in $b^{2} - 2 b + 1 - b^{2}$ .

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Step 3.2.2.1

Subtract $b^{2}$ from $b^{2}$ .

$- 2 b + 1 + 0 = - 13$

Step 3.2.2.2

Add $- 2 b + 1$ and $0$ .

$- 2 b + 1 = - 13$

Step 3.3

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

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Step 3.3.1

Subtract $1$ from both sides of the equation.

$- 2 b = - 13 - 1$

Step 3.3.2

Subtract $1$ from $- 13$ .

$- 2 b = - 14$

Step 3.4

Divide each term in $- 2 b = - 14$ by $- 2$ and simplify.

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Step 3.4.1

Divide each term in $- 2 b = - 14$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{- 14}{- 2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{- 14}{- 2}$

Step 3.4.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 14}{- 2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $- 14$ by $- 2$ .

$b = 7$