Algebra Examples

$\sqrt{2 b - 15} = \sqrt{\frac{b}{8}}$

Step 1

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

${\sqrt{2 b - 15}}^{2} = {\sqrt{\frac{b}{8}}}^{2}$

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 $\sqrt{2 b - 15}$ as ${(2 b - 15)}^{\frac{1}{2}}$ .

${({(2 b - 15)}^{\frac{1}{2}})}^{2} = {\sqrt{\frac{b}{8}}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(2 b - 15)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(2 b - 15)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(2 b - 15)}^{\frac{1}{2} \cdot 2} = {\sqrt{\frac{b}{8}}}^{2}$

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.

${(2 b - 15)}^{\frac{1}{2} \cdot 2} = {\sqrt{\frac{b}{8}}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 b - 15)}^{1} = {\sqrt{\frac{b}{8}}}^{2}$

Step 2.2.1.2

Simplify.

$2 b - 15 = {\sqrt{\frac{b}{8}}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{\frac{b}{8}}}^{2}$ .

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Step 2.3.1.1

Rewrite $\frac{b}{8}$ as ${(\frac{1}{2})}^{2} \frac{b}{2}$ .

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Step 2.3.1.1.1

Factor the perfect power $1^{2}$ out of $b$ .

$2 b - 15 = {\sqrt{\frac{1^{2} b}{8}}}^{2}$

Step 2.3.1.1.2

Factor the perfect power $2^{2}$ out of $8$ .

$2 b - 15 = {\sqrt{\frac{1^{2} b}{2^{2} \cdot 2}}}^{2}$

Step 2.3.1.1.3

Rearrange the fraction $\frac{1^{2} b}{2^{2} \cdot 2}$ .

$2 b - 15 = {\sqrt{{(\frac{1}{2})}^{2} \frac{b}{2}}}^{2}$

Step 2.3.1.2

Pull terms out from under the radical.

$2 b - 15 = {(\frac{1}{2} \sqrt{\frac{b}{2}})}^{2}$

Step 2.3.1.3

Rewrite $\sqrt{\frac{b}{2}}$ as $\frac{\sqrt{b}}{\sqrt{2}}$ .

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b}}{\sqrt{2}})}^{2}$

Step 2.3.1.4

Multiply $\frac{\sqrt{b}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$2 b - 15 = {(\frac{1}{2} (\frac{\sqrt{b}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}))}^{2}$

Step 2.3.1.5

Combine and simplify the denominator.

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Step 2.3.1.5.1

Multiply $\frac{\sqrt{b}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b} \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2}$

Step 2.3.1.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2}$

Step 2.3.1.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2}$

Step 2.3.1.5.4

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

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b} \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2}$

Step 2.3.1.5.5

Add $1$ and $1$ .

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b} \sqrt{2}}{{\sqrt{2}}^{2}})}^{2}$

Step 2.3.1.5.6

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

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Step 2.3.1.5.6.1

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

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2}$

Step 2.3.1.5.6.2

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

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2}$

Step 2.3.1.5.6.3

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

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b} \sqrt{2}}{2^{\frac{2}{2}}})}^{2}$

Step 2.3.1.5.6.4

Cancel the common factor of $2$ .

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Step 2.3.1.5.6.4.1

Cancel the common factor.

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b} \sqrt{2}}{2^{\frac{2}{2}}})}^{2}$

Step 2.3.1.5.6.4.2

Rewrite the expression.

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b} \sqrt{2}}{2^{1}})}^{2}$

Step 2.3.1.5.6.5

Evaluate the exponent.

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b} \sqrt{2}}{2})}^{2}$

Step 2.3.1.6

Combine using the product rule for radicals.

$2 b - 15 = {(\frac{1}{2} \cdot \frac{\sqrt{b \cdot 2}}{2})}^{2}$

Step 2.3.1.7

Multiply $\frac{1}{2} \cdot \frac{\sqrt{b \cdot 2}}{2}$ .

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Step 2.3.1.7.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{b \cdot 2}}{2}$ .

$2 b - 15 = {(\frac{\sqrt{b \cdot 2}}{2 \cdot 2})}^{2}$

Step 2.3.1.7.2

Multiply $2$ by $2$ .

$2 b - 15 = {(\frac{\sqrt{b \cdot 2}}{4})}^{2}$

Step 2.3.1.8

Apply the product rule to $\frac{\sqrt{b \cdot 2}}{4}$ .

$2 b - 15 = \frac{{\sqrt{b \cdot 2}}^{2}}{4^{2}}$

Step 2.3.1.9

Simplify the numerator.

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Step 2.3.1.9.1

Rewrite ${\sqrt{b \cdot 2}}^{2}$ as $b \cdot 2$ .

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Step 2.3.1.9.1.1

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

$2 b - 15 = \frac{{({(b \cdot 2)}^{\frac{1}{2}})}^{2}}{4^{2}}$

Step 2.3.1.9.1.2

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

$2 b - 15 = \frac{{(b \cdot 2)}^{\frac{1}{2} \cdot 2}}{4^{2}}$

Step 2.3.1.9.1.3

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

$2 b - 15 = \frac{{(b \cdot 2)}^{\frac{2}{2}}}{4^{2}}$

Step 2.3.1.9.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.9.1.4.1

Cancel the common factor.

$2 b - 15 = \frac{{(b \cdot 2)}^{\frac{2}{2}}}{4^{2}}$

Step 2.3.1.9.1.4.2

Rewrite the expression.

$2 b - 15 = \frac{{(b \cdot 2)}^{1}}{4^{2}}$

Step 2.3.1.9.1.5

Simplify.

$2 b - 15 = \frac{b \cdot 2}{4^{2}}$

Step 2.3.1.9.2

Move $2$ to the left of $b$ .

$2 b - 15 = \frac{2 b}{4^{2}}$

Step 2.3.1.10

Reduce the expression by cancelling the common factors.

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Step 2.3.1.10.1

Raise $4$ to the power of $2$ .

$2 b - 15 = \frac{2 b}{16}$

Step 2.3.1.10.2

Cancel the common factor of $2$ and $16$ .

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Step 2.3.1.10.2.1

Factor $2$ out of $2 b$ .

$2 b - 15 = \frac{2 (b)}{16}$

Step 2.3.1.10.2.2

Cancel the common factors.

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Step 2.3.1.10.2.2.1

Factor $2$ out of $16$ .

$2 b - 15 = \frac{2 b}{2 \cdot 8}$

Step 2.3.1.10.2.2.2

Cancel the common factor.

$2 b - 15 = \frac{2 b}{2 \cdot 8}$

Step 2.3.1.10.2.2.3

Rewrite the expression.

$2 b - 15 = \frac{b}{8}$

Step 3

Solve for $b$ .

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Step 3.1

Multiply both sides by $8$ .

$(2 b - 15) \cdot 8 = \frac{b}{8} \cdot 8$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $(2 b - 15) \cdot 8$ .

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Step 3.2.1.1.1

Apply the distributive property.

$2 b \cdot 8 - 15 \cdot 8 = \frac{b}{8} \cdot 8$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $8$ by $2$ .

$16 b - 15 \cdot 8 = \frac{b}{8} \cdot 8$

Step 3.2.1.1.2.2

Multiply $- 15$ by $8$ .

$16 b - 120 = \frac{b}{8} \cdot 8$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Cancel the common factor of $8$ .

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Step 3.2.2.1.1

Cancel the common factor.

$16 b - 120 = \frac{b}{8} \cdot 8$

Step 3.2.2.1.2

Rewrite the expression.

$16 b - 120 = b$

Step 3.3

Solve for $b$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $b$ from both sides of the equation.

$16 b - 120 - b = 0$

Step 3.3.1.2

Subtract $b$ from $16 b$ .

$15 b - 120 = 0$

Step 3.3.2

Add $120$ to both sides of the equation.

$15 b = 120$

Step 3.3.3

Divide each term in $15 b = 120$ by $15$ and simplify.

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Step 3.3.3.1

Divide each term in $15 b = 120$ by $15$ .

$\frac{15 b}{15} = \frac{120}{15}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $15$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{15 b}{15} = \frac{120}{15}$

Step 3.3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{120}{15}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $120$ by $15$ .

$b = 8$