Algebra Examples

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

Step 1

Move $4^{\frac{x}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

Rewrite $8$ as $2^{3}$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

$2^{\frac{3 x}{2}} \cdot {(2^{2})}^{- \frac{x}{3}} = 2^{- \frac{5}{2}}$

Step 5

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

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Step 5.1

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

$2^{\frac{3 x}{2}} \cdot 2^{2 (- \frac{x}{3})} = 2^{- \frac{5}{2}}$

Step 5.2

Multiply $2 (- \frac{x}{3})$ .

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Step 5.2.1

Multiply $- 1$ by $2$ .

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

Step 5.2.2

Combine $- 2$ and $\frac{x}{3}$ .

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

Step 5.3

Move the negative in front of the fraction.

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

Step 6

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

$2^{\frac{3 x}{2} - \frac{2 x}{3}} = 2^{- \frac{5}{2}}$

Step 7

To write $\frac{3 x}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 8

To write $- \frac{2 x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 9

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{3 x}{2}$ by $\frac{3}{3}$ .

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

Step 9.2

Multiply $2$ by $3$ .

$2^{\frac{3 x \cdot 3}{6} - \frac{2 x}{3} \cdot \frac{2}{2}} = 2^{- \frac{5}{2}}$

Step 9.3

Multiply $\frac{2 x}{3}$ by $\frac{2}{2}$ .

$2^{\frac{3 x \cdot 3}{6} - \frac{2 x \cdot 2}{3 \cdot 2}} = 2^{- \frac{5}{2}}$

Step 9.4

Multiply $3$ by $2$ .

$2^{\frac{3 x \cdot 3}{6} - \frac{2 x \cdot 2}{6}} = 2^{- \frac{5}{2}}$

Step 10

Combine the numerators over the common denominator.

$2^{\frac{3 x \cdot 3 - 2 x \cdot 2}{6}} = 2^{- \frac{5}{2}}$

Step 11

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

$\frac{3 x \cdot 3 - 2 x \cdot 2}{6} = - \frac{5}{2}$

Step 12

Solve for $x$ .

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Step 12.1

Multiply both sides by $6$ .

$\frac{3 x \cdot 3 - 2 x \cdot 2}{6} \cdot 6 = - \frac{5}{2} \cdot 6$

Step 12.2

Simplify.

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Step 12.2.1

Simplify the left side.

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Step 12.2.1.1

Simplify $\frac{3 x \cdot 3 - 2 x \cdot 2}{6} \cdot 6$ .

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Step 12.2.1.1.1

Simplify the numerator.

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Step 12.2.1.1.1.1

Factor $x$ out of $3 x \cdot 3 - 2 x \cdot 2$ .

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Step 12.2.1.1.1.1.1

Factor $x$ out of $3 x \cdot 3$ .

$\frac{x (3 \cdot 3) - 2 x \cdot 2}{6} \cdot 6 = - \frac{5}{2} \cdot 6$

Step 12.2.1.1.1.1.2

Factor $x$ out of $- 2 x \cdot 2$ .

$\frac{x (3 \cdot 3) + x (- 2 \cdot 2)}{6} \cdot 6 = - \frac{5}{2} \cdot 6$

Step 12.2.1.1.1.1.3

Factor $x$ out of $x (3 \cdot 3) + x (- 2 \cdot 2)$ .

$\frac{x (3 \cdot 3 - 2 \cdot 2)}{6} \cdot 6 = - \frac{5}{2} \cdot 6$

Step 12.2.1.1.1.2

Multiply $3$ by $3$ .

$\frac{x (9 - 2 \cdot 2)}{6} \cdot 6 = - \frac{5}{2} \cdot 6$

Step 12.2.1.1.1.3

Multiply $- 2$ by $2$ .

$\frac{x (9 - 4)}{6} \cdot 6 = - \frac{5}{2} \cdot 6$

Step 12.2.1.1.1.4

Subtract $4$ from $9$ .

$\frac{x \cdot 5}{6} \cdot 6 = - \frac{5}{2} \cdot 6$

Step 12.2.1.1.2

Reduce the expression by cancelling the common factors.

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Step 12.2.1.1.2.1

Cancel the common factor of $6$ .

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Step 12.2.1.1.2.1.1

Cancel the common factor.

$\frac{x \cdot 5}{6} \cdot 6 = - \frac{5}{2} \cdot 6$

Step 12.2.1.1.2.1.2

Rewrite the expression.

$x \cdot 5 = - \frac{5}{2} \cdot 6$

Step 12.2.1.1.2.2

Move $5$ to the left of $x$ .

$5 x = - \frac{5}{2} \cdot 6$

Step 12.2.2

Simplify the right side.

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Step 12.2.2.1

Simplify $- \frac{5}{2} \cdot 6$ .

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Step 12.2.2.1.1

Cancel the common factor of $2$ .

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Step 12.2.2.1.1.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$5 x = \frac{- 5}{2} \cdot 6$

Step 12.2.2.1.1.2

Factor $2$ out of $6$ .

$5 x = \frac{- 5}{2} \cdot (2 (3))$

Step 12.2.2.1.1.3

Cancel the common factor.

$5 x = \frac{- 5}{2} \cdot (2 \cdot 3)$

Step 12.2.2.1.1.4

Rewrite the expression.

$5 x = - 5 \cdot 3$

Step 12.2.2.1.2

Multiply $- 5$ by $3$ .

$5 x = - 15$

Step 12.3

Divide each term in $5 x = - 15$ by $5$ and simplify.

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Step 12.3.1

Divide each term in $5 x = - 15$ by $5$ .

$\frac{5 x}{5} = \frac{- 15}{5}$

Step 12.3.2

Simplify the left side.

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Step 12.3.2.1

Cancel the common factor of $5$ .

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Step 12.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 15}{5}$

Step 12.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 15}{5}$

Step 12.3.3

Simplify the right side.

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Step 12.3.3.1

Divide $- 15$ by $5$ .

$x = - 3$