Algebra Examples

${(\frac{1}{3})}^{d - 5} = 81$

Step 1

Apply the product rule to $\frac{1}{3}$ .

$\frac{1^{d - 5}}{3^{d - 5}} = 81$

Step 2

One to any power is one.

$\frac{1}{3^{d - 5}} = 81$

Step 3

Move $3^{d - 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$3^{- (d - 5)} = 81$

Step 4

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

$3^{- (d - 5)} = 3^{4}$

Step 5

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

$- (d - 5) = 4$

Step 6

Solve for $d$ .

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Step 6.1

Simplify $- (d - 5)$ .

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Step 6.1.1

Apply the distributive property.

$- d - - 5 = 4$

Step 6.1.2

Multiply $- 1$ by $- 5$ .

$- d + 5 = 4$

Step 6.2

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

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Step 6.2.1

Subtract $5$ from both sides of the equation.

$- d = 4 - 5$

Step 6.2.2

Subtract $5$ from $4$ .

$- d = - 1$

Step 6.3

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

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Step 6.3.1

Divide each term in $- d = - 1$ by $- 1$ .

$\frac{- d}{- 1} = \frac{- 1}{- 1}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Dividing two negative values results in a positive value.

$\frac{d}{1} = \frac{- 1}{- 1}$

Step 6.3.2.2

Divide $d$ by $1$ .

$d = \frac{- 1}{- 1}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $- 1$ by $- 1$ .

$d = 1$