Algebra Examples

$4 b^{- \frac{3}{4}} + 10 = \frac{21}{2}$

Step 1

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

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Step 1.1

Subtract $10$ from both sides of the equation.

$4 b^{- \frac{3}{4}} = \frac{21}{2} - 10$

Step 1.2

To write $- 10$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 b^{- \frac{3}{4}} = \frac{21}{2} - 10 \cdot \frac{2}{2}$

Step 1.3

Combine $- 10$ and $\frac{2}{2}$ .

$4 b^{- \frac{3}{4}} = \frac{21}{2} + \frac{- 10 \cdot 2}{2}$

Step 1.4

Combine the numerators over the common denominator.

$4 b^{- \frac{3}{4}} = \frac{21 - 10 \cdot 2}{2}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 10$ by $2$ .

$4 b^{- \frac{3}{4}} = \frac{21 - 20}{2}$

Step 1.5.2

Subtract $20$ from $21$ .

$4 b^{- \frac{3}{4}} = \frac{1}{2}$

Step 2

Raise each side of the equation to the power of $- \frac{4}{3}$ to eliminate the fractional exponent on the left side.

${(4 b^{- \frac{3}{4}})}^{- \frac{4}{3}} = {(\frac{1}{2})}^{- \frac{4}{3}}$

Step 3

Simplify the exponent.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify ${(4 b^{- \frac{3}{4}})}^{- \frac{4}{3}}$ .

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Step 3.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(4 \frac{1}{b^{\frac{3}{4}}})}^{- \frac{4}{3}} = {(\frac{1}{2})}^{- \frac{4}{3}}$

Step 3.1.1.2

Combine $4$ and $\frac{1}{b^{\frac{3}{4}}}$ .

${(\frac{4}{b^{\frac{3}{4}}})}^{- \frac{4}{3}} = {(\frac{1}{2})}^{- \frac{4}{3}}$

Step 3.1.1.3

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{b^{\frac{3}{4}}}{4})}^{\frac{4}{3}} = {(\frac{1}{2})}^{- \frac{4}{3}}$

Step 3.1.1.4

Apply the product rule to $\frac{b^{\frac{3}{4}}}{4}$ .

$\frac{{(b^{\frac{3}{4}})}^{\frac{4}{3}}}{4^{\frac{4}{3}}} = {(\frac{1}{2})}^{- \frac{4}{3}}$

Step 3.1.1.5

Simplify the numerator.

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Step 3.1.1.5.1

Multiply the exponents in ${(b^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 3.1.1.5.1.1

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

$\frac{b^{\frac{3}{4} \cdot \frac{4}{3}}}{4^{\frac{4}{3}}} = {(\frac{1}{2})}^{- \frac{4}{3}}$

Step 3.1.1.5.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.5.1.2.1

Cancel the common factor.

$\frac{b^{\frac{3}{4} \cdot \frac{4}{3}}}{4^{\frac{4}{3}}} = {(\frac{1}{2})}^{- \frac{4}{3}}$

Step 3.1.1.5.1.2.2

Rewrite the expression.

$\frac{b^{\frac{1}{4} \cdot 4}}{4^{\frac{4}{3}}} = {(\frac{1}{2})}^{- \frac{4}{3}}$

Step 3.1.1.5.1.3

Cancel the common factor of $4$ .

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Step 3.1.1.5.1.3.1

Cancel the common factor.

$\frac{b^{\frac{1}{4} \cdot 4}}{4^{\frac{4}{3}}} = {(\frac{1}{2})}^{- \frac{4}{3}}$

Step 3.1.1.5.1.3.2

Rewrite the expression.

$\frac{b^{1}}{4^{\frac{4}{3}}} = {(\frac{1}{2})}^{- \frac{4}{3}}$

Step 3.1.1.5.2

Simplify.

$\frac{b}{4^{\frac{4}{3}}} = {(\frac{1}{2})}^{- \frac{4}{3}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{b}{4^{\frac{4}{3}}} = 2^{\frac{4}{3}}$

Step 4

Solve for $b$ .

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Step 4.1

Multiply both sides of the equation by $4^{\frac{4}{3}}$ .

$4^{\frac{4}{3}} \frac{b}{4^{\frac{4}{3}}} = 4^{\frac{4}{3}} \cdot 2^{\frac{4}{3}}$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $4^{\frac{4}{3}}$ .

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Step 4.2.1.1.1

Cancel the common factor.

$4^{\frac{4}{3}} \frac{b}{4^{\frac{4}{3}}} = 4^{\frac{4}{3}} \cdot 2^{\frac{4}{3}}$

Step 4.2.1.1.2

Rewrite the expression.

$b = 4^{\frac{4}{3}} \cdot 2^{\frac{4}{3}}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $4^{\frac{4}{3}} \cdot 2^{\frac{4}{3}}$ .

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Step 4.2.2.1.1

Multiply $4^{\frac{4}{3}} \cdot 2^{\frac{4}{3}}$ .

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Step 4.2.2.1.1.1

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

$b = {(2^{2})}^{\frac{4}{3}} \cdot 2^{\frac{4}{3}}$

Step 4.2.2.1.1.2

Multiply the exponents in ${(2^{2})}^{\frac{4}{3}}$ .

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Step 4.2.2.1.1.2.1

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

$b = 2^{2 (\frac{4}{3})} \cdot 2^{\frac{4}{3}}$

Step 4.2.2.1.1.2.2

Multiply $2 (\frac{4}{3})$ .

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Step 4.2.2.1.1.2.2.1

Combine $2$ and $\frac{4}{3}$ .

$b = 2^{\frac{2 \cdot 4}{3}} \cdot 2^{\frac{4}{3}}$

Step 4.2.2.1.1.2.2.2

Multiply $2$ by $4$ .

$b = 2^{\frac{8}{3}} \cdot 2^{\frac{4}{3}}$

Step 4.2.2.1.1.3

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

$b = 2^{\frac{8}{3} + \frac{4}{3}}$

Step 4.2.2.1.1.4

Combine the numerators over the common denominator.

$b = 2^{\frac{8 + 4}{3}}$

Step 4.2.2.1.1.5

Add $8$ and $4$ .

$b = 2^{\frac{12}{3}}$

Step 4.2.2.1.1.6

Cancel the common factor of $12$ and $3$ .

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Step 4.2.2.1.1.6.1

Factor $3$ out of $12$ .

$b = 2^{\frac{3 \cdot 4}{3}}$

Step 4.2.2.1.1.6.2

Cancel the common factors.

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Step 4.2.2.1.1.6.2.1

Factor $3$ out of $3$ .

$b = 2^{\frac{3 \cdot 4}{3 (1)}}$

Step 4.2.2.1.1.6.2.2

Cancel the common factor.

$b = 2^{\frac{3 \cdot 4}{3 \cdot 1}}$

Step 4.2.2.1.1.6.2.3

Rewrite the expression.

$b = 2^{\frac{4}{1}}$

Step 4.2.2.1.1.6.2.4

Divide $4$ by $1$ .

$b = 2^{4}$

Step 4.2.2.1.2

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

$b = 16$