Solve [frac{ab^{-1}}{a^{-3}b^{2}}]^{-5}div[a^-2b/a^3b^-4]^3

$$[ \frac { a b ^ { - 1 } } { a ^ { - 3 } b ^ { 2 } } ] ^ { - 5 } \div [ \frac { a ^ { - 2 } b } { a ^ { 3 } b ^ { - 4 } } ] ^ { 3 }$$

Evaluate

$$\frac{1}{a^{5}}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\left(\frac{\frac{1}{b}a^{4}}{b^{2}}\right)^{-5}}{\left(\frac{a^{-2}b}{a^{3}b^{-4}}\right)^{3}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\left(\frac{a^{4}}{b^{3}}\right)^{-5}}{\left(\frac{a^{-2}b}{a^{3}b^{-4}}\right)^{3}}$$

To raise \frac{a^{4}}{b^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(a^{4}\right)^{-5}}{\left(b^{3}\right)^{-5}}}{\left(\frac{a^{-2}b}{a^{3}b^{-4}}\right)^{3}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\frac{\left(a^{4}\right)^{-5}}{\left(b^{3}\right)^{-5}}}{\left(\frac{a^{-2}b^{5}}{a^{3}}\right)^{3}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\frac{\left(a^{4}\right)^{-5}}{\left(b^{3}\right)^{-5}}}{\left(\frac{b^{5}}{a^{5}}\right)^{3}}$$

To raise \frac{b^{5}}{a^{5}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(a^{4}\right)^{-5}}{\left(b^{3}\right)^{-5}}}{\frac{\left(b^{5}\right)^{3}}{\left(a^{5}\right)^{3}}}$$

Divide \frac{\left(a^{4}\right)^{-5}}{\left(b^{3}\right)^{-5}} by \frac{\left(b^{5}\right)^{3}}{\left(a^{5}\right)^{3}} by multiplying \frac{\left(a^{4}\right)^{-5}}{\left(b^{3}\right)^{-5}} by the reciprocal of \frac{\left(b^{5}\right)^{3}}{\left(a^{5}\right)^{3}}.

$$\frac{\left(a^{4}\right)^{-5}\left(a^{5}\right)^{3}}{\left(b^{3}\right)^{-5}\left(b^{5}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -5 to get -20.

$$\frac{a^{-20}\left(a^{5}\right)^{3}}{\left(b^{3}\right)^{-5}\left(b^{5}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{a^{-20}a^{15}}{\left(b^{3}\right)^{-5}\left(b^{5}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add -20 and 15 to get -5.

$$\frac{a^{-5}}{\left(b^{3}\right)^{-5}\left(b^{5}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -5 to get -15.

$$\frac{a^{-5}}{b^{-15}\left(b^{5}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{a^{-5}}{b^{-15}b^{15}}$$

Multiply b^{-15} and b^{15} to get 1.

$$\frac{a^{-5}}{1}$$

Anything divided by one gives itself.

$$a^{-5}$$

Expand

$$\frac{1}{a^{5}}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\left(\frac{\frac{1}{b}a^{4}}{b^{2}}\right)^{-5}}{\left(\frac{a^{-2}b}{a^{3}b^{-4}}\right)^{3}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\left(\frac{a^{4}}{b^{3}}\right)^{-5}}{\left(\frac{a^{-2}b}{a^{3}b^{-4}}\right)^{3}}$$

To raise \frac{a^{4}}{b^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(a^{4}\right)^{-5}}{\left(b^{3}\right)^{-5}}}{\left(\frac{a^{-2}b}{a^{3}b^{-4}}\right)^{3}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\frac{\left(a^{4}\right)^{-5}}{\left(b^{3}\right)^{-5}}}{\left(\frac{a^{-2}b^{5}}{a^{3}}\right)^{3}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\frac{\left(a^{4}\right)^{-5}}{\left(b^{3}\right)^{-5}}}{\left(\frac{b^{5}}{a^{5}}\right)^{3}}$$

To raise \frac{b^{5}}{a^{5}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(a^{4}\right)^{-5}}{\left(b^{3}\right)^{-5}}}{\frac{\left(b^{5}\right)^{3}}{\left(a^{5}\right)^{3}}}$$

Divide \frac{\left(a^{4}\right)^{-5}}{\left(b^{3}\right)^{-5}} by \frac{\left(b^{5}\right)^{3}}{\left(a^{5}\right)^{3}} by multiplying \frac{\left(a^{4}\right)^{-5}}{\left(b^{3}\right)^{-5}} by the reciprocal of \frac{\left(b^{5}\right)^{3}}{\left(a^{5}\right)^{3}}.

$$\frac{\left(a^{4}\right)^{-5}\left(a^{5}\right)^{3}}{\left(b^{3}\right)^{-5}\left(b^{5}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -5 to get -20.

$$\frac{a^{-20}\left(a^{5}\right)^{3}}{\left(b^{3}\right)^{-5}\left(b^{5}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{a^{-20}a^{15}}{\left(b^{3}\right)^{-5}\left(b^{5}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add -20 and 15 to get -5.

$$\frac{a^{-5}}{\left(b^{3}\right)^{-5}\left(b^{5}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -5 to get -15.

$$\frac{a^{-5}}{b^{-15}\left(b^{5}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{a^{-5}}{b^{-15}b^{15}}$$

Multiply b^{-15} and b^{15} to get 1.

$$\frac{a^{-5}}{1}$$

Anything divided by one gives itself.

$$a^{-5}$$