Simplify x^(-1/6)*x^(-6/7)

$x^{- \frac{1}{6}} \cdot x^{- \frac{6}{7}}$

Multiply $x^{- \frac{1}{6}}$ by $x^{- \frac{6}{7}}$ by adding the exponents.

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

$x^{- \frac{1}{6} - \frac{6}{7}}$

To write $- \frac{1}{6}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x^{- \frac{1}{6} \cdot \frac{7}{7} - \frac{6}{7}}$

To write $- \frac{6}{7}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x^{- \frac{1}{6} \cdot \frac{7}{7} - \frac{6}{7} \cdot \frac{6}{6}}$

Write each expression with a common denominator of $42$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{6}$ by $\frac{7}{7}$ .

$x^{- \frac{7}{6 \cdot 7} - \frac{6}{7} \cdot \frac{6}{6}}$

Multiply $6$ by $7$ .

$x^{- \frac{7}{42} - \frac{6}{7} \cdot \frac{6}{6}}$

Multiply $\frac{6}{7}$ by $\frac{6}{6}$ .

$x^{- \frac{7}{42} - \frac{6 \cdot 6}{7 \cdot 6}}$

Multiply $7$ by $6$ .

$x^{- \frac{7}{42} - \frac{6 \cdot 6}{42}}$

Combine the numerators over the common denominator.

$x^{\frac{- 7 - 6 \cdot 6}{42}}$

Simplify the numerator.

Multiply $- 6$ by $6$ .

$x^{\frac{- 7 - 36}{42}}$

Subtract $36$ from $- 7$ .

$x^{\frac{- 43}{42}}$

Move the negative in front of the fraction.

$x^{- \frac{43}{42}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{\frac{43}{42}}}$