Solve for x 9^(3x-10)=(1/81)^(1/6)

$9^{3 x - 10} = {(\frac{1}{81})}^{\frac{1}{6}}$

Step 1

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

$9^{3 x - 10} = \frac{1^{\frac{1}{6}}}{81^{\frac{1}{6}}}$

Step 2

One to any power is one.

$9^{3 x - 10} = \frac{1}{81^{\frac{1}{6}}}$

Step 3

Move $81^{\frac{1}{6}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$9^{3 x - 10} = 81^{- \frac{1}{6}}$

Step 4

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

$9^{3 x - 10} = 9^{2 (- \frac{1}{6})}$

Step 5

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

$3 x - 10 = 2 (- \frac{1}{6})$

Step 6

Solve for $x$ .

Simplify $2 (- \frac{1}{6})$ .

Rewrite.

$3 x - 10 = 0 + 0 + 2 (- \frac{1}{6})$

Simplify by adding zeros.

$3 x - 10 = 2 (- \frac{1}{6})$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$3 x - 10 = 2 (\frac{- 1}{6})$

Factor $2$ out of $6$ .

$3 x - 10 = 2 \frac{- 1}{2 (3)}$

Cancel the common factor.

$3 x - 10 = 2 \frac{- 1}{2 \cdot 3}$

Rewrite the expression.

$3 x - 10 = \frac{- 1}{3}$

Move the negative in front of the fraction.

$3 x - 10 = - \frac{1}{3}$

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

Add $10$ to both sides of the equation.

$3 x = - \frac{1}{3} + 10$

To write $10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 x = - \frac{1}{3} + 10 \cdot \frac{3}{3}$

Combine $10$ and $\frac{3}{3}$ .

$3 x = - \frac{1}{3} + \frac{10 \cdot 3}{3}$

Combine the numerators over the common denominator.

$3 x = \frac{- 1 + 10 \cdot 3}{3}$

Simplify the numerator.

Multiply $10$ by $3$ .

$3 x = \frac{- 1 + 30}{3}$

Add $- 1$ and $30$ .

$3 x = \frac{29}{3}$

Divide each term in $3 x = \frac{29}{3}$ by $3$ and simplify.

Divide each term in $3 x = \frac{29}{3}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{29}{3}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{29}{3}}{3}$

Divide $x$ by $1$ .

$x = \frac{\frac{29}{3}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{29}{3} \cdot \frac{1}{3}$

Multiply $\frac{29}{3} \cdot \frac{1}{3}$ .

Multiply $\frac{29}{3}$ by $\frac{1}{3}$ .

$x = \frac{29}{3 \cdot 3}$

Multiply $3$ by $3$ .

$x = \frac{29}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{29}{9}$

Decimal Form:

$x = 3.\overline{2}$

Mixed Number Form:

$x = 3 \frac{2}{9}$

Solve for x 9^(3x-10)=(1/81)^(1/6)

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