Rewrite the negative exponent as a fraction.

\(\displaystyle x^{-b} = \frac{1}{x^b}\)

Following this rule, rewrite the given expression.

\(\displaystyle (\frac{4}{5})^{-2}=\frac{1}{(\frac{4}{5})^2} = \frac{1}{\frac{4}{5} \cdot \frac{4}{5}}= \frac{1}{\frac{16}{25}}\)

Take the reciprocal of the denominator to eliminate the complex fraction.

The answer is:  \(\displaystyle \frac{25}{16}\)

To eliminate the negative exponent, we will need to convert the terms to a fraction.

\(\displaystyle x^{-a} = \frac{1}{x^a}\)

According to this rule, convert both terms.

\(\displaystyle 2^{-1}+3^{-2} = \frac{1}{2}+\frac{1}{3^2}\)

Simplify the fractions.

\(\displaystyle \frac{1}{2}+\frac{1}{9} = \frac{9}{18}+ \frac{2}{18} =\frac{11}{18}\)

The answer is:  \(\displaystyle \frac{11}{18}\)

Convert the numerator and denominator to fractions by using the formula below.

\(\displaystyle a^{-n} =\frac{1}{a^n}\)

\(\displaystyle \frac{6^{-3}}{8^{-3}} = \frac{\frac{1}{6^3}}{\frac{1}{8^3}}\)

Rewrite the fractions using a division sign.

\(\displaystyle \frac{\frac{1}{6^3}}{\frac{1}{8^3}} =\frac{1}{6^3} \div\frac{1}{8^3}\)

Take the reciprocal of the second term and change the division sign to a multiplication symbol.

\(\displaystyle \frac{1}{6^3} \times 8^3 = \frac{8^3}{6^3} = \frac{8\cdot 8\cdot 8}{6\cdot 6\cdot 6} =\frac{4\cdot 4\cdot 4}{3\cdot 3\cdot 3} = \frac{64}{27}\)

The answer is:  \(\displaystyle \frac{64}{27}\)

Write the rule for negative exponents.

\(\displaystyle x^{-a} =\frac{1}{x^a}\)

Convert the powers on the numerator and denominator.

\(\displaystyle \frac{2^{-1}+3^{-1}}{2^{-2}-3^{-2}} = \frac{\frac{1}{2}+\frac{1}{3}}{\frac{1}{4}-\frac{1}{9}}\)

Add the numerator.  Find the least common denominator and add the numerators.

\(\displaystyle \frac{1}{2}+\frac{1}{3} = \frac{3}{6}+\frac{2}{6} = \frac{5}{6}\)

Subtract the denominator also by converting to the least common denominator.

\(\displaystyle \frac{1}{4}-\frac{1}{9} = \frac{9}{36}-\frac{4}{36} = \frac{5}{36}\)

Divide the numerator with the denominator.  Convert the complex fraction by rewriting it using a division sign.  Swap the numerator and denominator of the second term and change the division sign to a multiplication sign.

\(\displaystyle \frac{\frac{5}{6}}{\frac{5}{36}} = \frac{5}{6}\div \frac{5}{36} = \frac{5}{6}\times \frac{36}{5} = 6\)

The answer is:  \(\displaystyle 6\)

Whenever we have a negative exponent, we can convert that to a fraction.

\(\displaystyle x^{-a} = \frac{1}{x^a}\)

Evaluate the inner term.

\(\displaystyle (\frac{3}{4^{-2}})^{-2} = (\frac{3}{\frac{1}{4^2}})^{-2} = (\frac{3}{\frac{1}{16}})^{-2}\)

The complex fraction can be reduced as follows:

\(\displaystyle \frac{3}{\frac{1}{16}} = 3\div \frac{1}{16} = 3\times \frac{16}{1} = 48\)

Substitute this value into the parentheses.  

\(\displaystyle (48)^{-2} = \frac{1}{48^2} = \frac{1}{2304}\)

The answer is:  \(\displaystyle \frac{1}{2304}\)

Convert the negative exponent as follows:

\(\displaystyle x^{-y}= \frac{1}{x^y}\)

Rewrite the expression without the negative exponent.

\(\displaystyle (\frac{2}{7})^{-2}=\frac{1}{(\frac{2}{7})^{2}}=\frac{1}{(\frac{4}{49})}\)

Rewrite the complex fraction using a division sign.

\(\displaystyle 1\div\frac{4}{49} = 1\times\frac{49}{4} = \frac{49}{4}\)

The answer is:  \(\displaystyle \frac{49}{4}\)

We evaluate negative exponents as 

\(\displaystyle x^{-a}=\frac{1}{x^a}\) 

which \(\displaystyle a\) is the positive exponent raising base \(\displaystyle x\).

Therefore 

\(\displaystyle 7^{-6}=\frac{1}{7^6}\).

We evaluate negative exponents as 

\(\displaystyle x^{-a}=\frac{1}{x^a}\) 

which \(\displaystyle a\) is the positive exponent raising base \(\displaystyle x\).

Therefore 

\(\displaystyle \frac{1}{3}^{-3}=\frac{1}{\frac{1}{3}^3}=\frac{1}{\frac{1}{27}}=27\)

The negative exponents can be rewritten as a fraction.

\(\displaystyle x^{-a} =\frac{1}{x^a}\)

Convert the terms.

\(\displaystyle 4^{-2}-2^{-6} = \frac{1}{4^2}-\frac{1}{2^6} = \frac{1}{16}- \frac{1}{64}\)

Convert the first fraction so that the denominator are similar.

\(\displaystyle \frac{1}{16}- \frac{1}{64} = \frac{1(4)}{16(4)}- \frac{1}{64} = \frac{4}{64}-\frac{1}{64}\)

Subtract the numerators.

The answer is:  \(\displaystyle \frac{3}{64}\)

To eliminate the negative exponent, use the following property:

\(\displaystyle x^{-a}=\frac{1}{x^a}\)

Rewrite the expression using this property.

\(\displaystyle (\frac{3}{5})^{-2} = \frac{1}{(\frac{3}{5})^2} = \frac{1}{(\frac{3}{5})(\frac{3}{5})} = \frac{1}{\frac{9}{25}}\)

Simplify the complex fraction.

\(\displaystyle \frac{1}{\frac{9}{25}} =1\div \frac{9}{25} =1\times \frac{25} {9} =\frac{25}{9}\)

The answer is:  \(\displaystyle \frac{25}{9}\)