Algebra Examples

$8^{\frac{1}{6}} \cdot 2^{x} = 32^{\frac{1}{2}}$

Step 1

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

${(2^{3})}^{\frac{1}{6}} \cdot 2^{x} = 32^{\frac{1}{2}}$

Step 2

Multiply the exponents in ${(2^{3})}^{\frac{1}{6}}$ .

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Step 2.1

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

$2^{3 (\frac{1}{6})} \cdot 2^{x} = 32^{\frac{1}{2}}$

Step 2.2

Cancel the common factor of $3$ .

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Step 2.2.1

Factor $3$ out of $6$ .

$2^{3 \frac{1}{3 (2)}} \cdot 2^{x} = 32^{\frac{1}{2}}$

Step 2.2.2

Cancel the common factor.

$2^{3 \frac{1}{3 \cdot 2}} \cdot 2^{x} = 32^{\frac{1}{2}}$

Step 2.2.3

Rewrite the expression.

$2^{\frac{1}{2}} \cdot 2^{x} = 32^{\frac{1}{2}}$

Step 3

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

$2^{\frac{1}{2} + x} = 32^{\frac{1}{2}}$

Step 4

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

$2^{\frac{1}{2} + x} = 2^{5 (\frac{1}{2})}$

Step 5

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

$\frac{1}{2} + x = 5 (\frac{1}{2})$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $5 (\frac{1}{2})$ .

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Step 6.1.1

Rewrite.

$\frac{1}{2} + x = 0 + 0 + 5 (\frac{1}{2})$

Step 6.1.2

Simplify by adding zeros.

$\frac{1}{2} + x = 5 (\frac{1}{2})$

Step 6.1.3

Combine $5$ and $\frac{1}{2}$ .

$\frac{1}{2} + x = \frac{5}{2}$

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 $\frac{1}{2}$ from both sides of the equation.

$x = \frac{5}{2} - \frac{1}{2}$

Step 6.2.2

Combine the numerators over the common denominator.

$x = \frac{5 - 1}{2}$

Step 6.2.3

Subtract $1$ from $5$ .

$x = \frac{4}{2}$

Step 6.2.4

Divide $4$ by $2$ .

$x = 2$