Algebra Examples

${(\frac{1}{81})}^{2 w} \cdot 27 = 81^{\frac{1}{2}}$

Step 1

Apply the product rule to $\frac{1}{81}$ .

$\frac{1^{2 w}}{81^{2 w}} \cdot 27 = 81^{\frac{1}{2}}$

Step 2

One to any power is one.

$\frac{1}{81^{2 w}} \cdot 27 = 81^{\frac{1}{2}}$

Step 3

Move $81^{2 w}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$81^{- 2 w} \cdot 27 = 81^{\frac{1}{2}}$

Step 4

Rewrite $81$ as $3^{4}$ .

${(3^{4})}^{- 2 w} \cdot 27 = 81^{\frac{1}{2}}$

Step 5

Multiply the exponents in ${(3^{4})}^{- 2 w}$ .

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Step 5.1

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

$3^{4 (- 2 w)} \cdot 27 = 81^{\frac{1}{2}}$

Step 5.2

Multiply $- 2$ by $4$ .

$3^{- 8 w} \cdot 27 = 81^{\frac{1}{2}}$

Step 6

Rewrite $27$ as $3^{3}$ .

$3^{- 8 w} \cdot 3^{3} = 81^{\frac{1}{2}}$

Step 7

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

$3^{- 8 w + 3} = 81^{\frac{1}{2}}$

Step 8

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

$3^{- 8 w + 3} = 3^{4 (\frac{1}{2})}$

Step 9

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

$- 8 w + 3 = 4 (\frac{1}{2})$

Step 10

Solve for $w$ .

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Step 10.1

Simplify $4 (\frac{1}{2})$ .

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Step 10.1.1

Rewrite.

$- 8 w + 3 = 0 + 0 + 4 (\frac{1}{2})$

Step 10.1.2

Simplify by adding zeros.

$- 8 w + 3 = 4 (\frac{1}{2})$

Step 10.1.3

Cancel the common factor of $2$ .

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Step 10.1.3.1

Factor $2$ out of $4$ .

$- 8 w + 3 = 2 (2) \frac{1}{2}$

Step 10.1.3.2

Cancel the common factor.

$- 8 w + 3 = 2 \cdot 2 \frac{1}{2}$

Step 10.1.3.3

Rewrite the expression.

$- 8 w + 3 = 2$

Step 10.2

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

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Step 10.2.1

Subtract $3$ from both sides of the equation.

$- 8 w = 2 - 3$

Step 10.2.2

Subtract $3$ from $2$ .

$- 8 w = - 1$

Step 10.3

Divide each term in $- 8 w = - 1$ by $- 8$ and simplify.

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Step 10.3.1

Divide each term in $- 8 w = - 1$ by $- 8$ .

$\frac{- 8 w}{- 8} = \frac{- 1}{- 8}$

Step 10.3.2

Simplify the left side.

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Step 10.3.2.1

Cancel the common factor of $- 8$ .

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Step 10.3.2.1.1

Cancel the common factor.

$\frac{- 8 w}{- 8} = \frac{- 1}{- 8}$

Step 10.3.2.1.2

Divide $w$ by $1$ .

$w = \frac{- 1}{- 8}$

Step 10.3.3

Simplify the right side.

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Step 10.3.3.1

Dividing two negative values results in a positive value.

$w = \frac{1}{8}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$w = \frac{1}{8}$

Decimal Form:

$w = 0.125$