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

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

Evaluate

$$a^{25}$$

Solution Steps

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

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

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

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

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

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

Cancel out a in both numerator and denominator.

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

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

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

To raise \frac{1}{a^{2}b^{3}} to a power, raise both numerator and denominator to the power and then divide.

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

Divide \frac{\left(b^{5}\right)^{-3}}{\left(a^{5}\right)^{-3}} by \frac{1^{5}}{\left(a^{2}b^{3}\right)^{5}} by multiplying \frac{\left(b^{5}\right)^{-3}}{\left(a^{5}\right)^{-3}} by the reciprocal of \frac{1^{5}}{\left(a^{2}b^{3}\right)^{5}}.

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

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

$$\frac{b^{-15}\left(a^{2}b^{3}\right)^{5}}{\left(a^{5}\right)^{-3}\times 1^{5}}$$

Expand \left(a^{2}b^{3}\right)^{5}.

$$\frac{b^{-15}\left(a^{2}\right)^{5}\left(b^{3}\right)^{5}}{\left(a^{5}\right)^{-3}\times 1^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{b^{-15}a^{10}\left(b^{3}\right)^{5}}{\left(a^{5}\right)^{-3}\times 1^{5}}$$

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

$$\frac{b^{-15}a^{10}b^{15}}{\left(a^{5}\right)^{-3}\times 1^{5}}$$

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

$$\frac{a^{10}}{\left(a^{5}\right)^{-3}\times 1^{5}}$$

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

$$\frac{a^{10}}{a^{-15}\times 1^{5}}$$

Calculate 1 to the power of 5 and get 1.

$$\frac{a^{10}}{a^{-15}\times 1}$$

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

$$a^{25}$$

Expand

$$a^{25}$$

Solution Steps

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

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

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

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

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

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

Cancel out a in both numerator and denominator.

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

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

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

To raise \frac{1}{a^{2}b^{3}} to a power, raise both numerator and denominator to the power and then divide.

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

Divide \frac{\left(b^{5}\right)^{-3}}{\left(a^{5}\right)^{-3}} by \frac{1^{5}}{\left(a^{2}b^{3}\right)^{5}} by multiplying \frac{\left(b^{5}\right)^{-3}}{\left(a^{5}\right)^{-3}} by the reciprocal of \frac{1^{5}}{\left(a^{2}b^{3}\right)^{5}}.

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

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

$$\frac{b^{-15}\left(a^{2}b^{3}\right)^{5}}{\left(a^{5}\right)^{-3}\times 1^{5}}$$

Expand \left(a^{2}b^{3}\right)^{5}.

$$\frac{b^{-15}\left(a^{2}\right)^{5}\left(b^{3}\right)^{5}}{\left(a^{5}\right)^{-3}\times 1^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{b^{-15}a^{10}\left(b^{3}\right)^{5}}{\left(a^{5}\right)^{-3}\times 1^{5}}$$

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

$$\frac{b^{-15}a^{10}b^{15}}{\left(a^{5}\right)^{-3}\times 1^{5}}$$

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

$$\frac{a^{10}}{\left(a^{5}\right)^{-3}\times 1^{5}}$$

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

$$\frac{a^{10}}{a^{-15}\times 1^{5}}$$

Calculate 1 to the power of 5 and get 1.

$$\frac{a^{10}}{a^{-15}\times 1}$$

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

$$a^{25}$$