Precalculus Examples

${(\frac{1}{3})}^{X} = 27$

Step 1

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

$\frac{1^{X}}{3^{X}} = 27$

Step 2

One to any power is one.

$\frac{1}{3^{X}} = 27$

Step 3

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

$3^{- X} = 27$

Step 4

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

$3^{- X} = 3^{3}$

Step 5

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

$- X = 3$

Step 6

Divide each term in $- X = 3$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- X = 3$ by $- 1$ .

$\frac{- X}{- 1} = \frac{3}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{X}{1} = \frac{3}{- 1}$

Step 6.2.2

Divide $X$ by $1$ .

$X = \frac{3}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $3$ by $- 1$ .

$X = - 3$