Algebra Examples

$16^{\frac{1}{5}} \cdot 2^{x} = 8^{\frac{3}{4}}$

Step 1

Rewrite $16$ as $2^{4}$ .

${(2^{4})}^{\frac{1}{5}} \cdot 2^{x} = 8^{\frac{3}{4}}$

Step 2

Multiply the exponents in ${(2^{4})}^{\frac{1}{5}}$ .

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Step 2.1

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

$2^{4 (\frac{1}{5})} \cdot 2^{x} = 8^{\frac{3}{4}}$

Step 2.2

Combine $4$ and $\frac{1}{5}$ .

$2^{\frac{4}{5}} \cdot 2^{x} = 8^{\frac{3}{4}}$

Step 3

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

$2^{\frac{4}{5} + x} = 8^{\frac{3}{4}}$

Step 4

Create equivalent expressions in the equation that all have equal bases.

$2^{\frac{4}{5} + x} = 2^{3 (\frac{3}{4})}$

Step 5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$\frac{4}{5} + x = 3 (\frac{3}{4})$

Step 6

Solve for $x$ .

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Step 6.1

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

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Step 6.1.1

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

$\frac{4}{5} + x = \frac{3 \cdot 3}{4}$

Step 6.1.2

Multiply $3$ by $3$ .

$\frac{4}{5} + x = \frac{9}{4}$

Step 6.2

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

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Step 6.2.1

Subtract $\frac{4}{5}$ from both sides of the equation.

$x = \frac{9}{4} - \frac{4}{5}$

Step 6.2.2

To write $\frac{9}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = \frac{9}{4} \cdot \frac{5}{5} - \frac{4}{5}$

Step 6.2.3

To write $- \frac{4}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{9}{4} \cdot \frac{5}{5} - \frac{4}{5} \cdot \frac{4}{4}$

Step 6.2.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.4.1

Multiply $\frac{9}{4}$ by $\frac{5}{5}$ .

$x = \frac{9 \cdot 5}{4 \cdot 5} - \frac{4}{5} \cdot \frac{4}{4}$

Step 6.2.4.2

Multiply $4$ by $5$ .

$x = \frac{9 \cdot 5}{20} - \frac{4}{5} \cdot \frac{4}{4}$

Step 6.2.4.3

Multiply $\frac{4}{5}$ by $\frac{4}{4}$ .

$x = \frac{9 \cdot 5}{20} - \frac{4 \cdot 4}{5 \cdot 4}$

Step 6.2.4.4

Multiply $5$ by $4$ .

$x = \frac{9 \cdot 5}{20} - \frac{4 \cdot 4}{20}$

Step 6.2.5

Combine the numerators over the common denominator.

$x = \frac{9 \cdot 5 - 4 \cdot 4}{20}$

Step 6.2.6

Simplify the numerator.

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Step 6.2.6.1

Multiply $9$ by $5$ .

$x = \frac{45 - 4 \cdot 4}{20}$

Step 6.2.6.2

Multiply $- 4$ by $4$ .

$x = \frac{45 - 16}{20}$

Step 6.2.6.3

Subtract $16$ from $45$ .

$x = \frac{29}{20}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{29}{20}$

Decimal Form:

$x = 1.45$

Mixed Number Form:

$x = 1 \frac{9}{20}$