Solve frac{a*b^{-2}*(a^{-1}*b^{2})^{4}*(a*b^{-1})^{2}}{a^-3*b*(a^2*b^-1)*(a^-1*b)}

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

Evaluate

$$ab^{3}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -3 and 2 to get -1.

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

To multiply powers of the same base, add their exponents. Add -1 and -1 to get -2.

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

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

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

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

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

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

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

Express \frac{1}{a}b^{2} as a single fraction.

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

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

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

Express \frac{1}{b}a as a single fraction.

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

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

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

Multiply \frac{\left(b^{2}\right)^{4}}{a^{4}} times \frac{a^{2}}{b^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out a^{2} in both numerator and denominator.

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

Express \frac{\left(b^{2}\right)^{4}}{a^{2}b^{2}}a^{3} as a single fraction.

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

Cancel out a^{2} in both numerator and denominator.

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

Express \frac{\frac{a\left(b^{2}\right)^{4}}{b^{2}}}{b^{3}} as a single fraction.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{ab^{8}}{b^{2}b^{3}}$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{ab^{8}}{b^{5}}$$

Cancel out b^{5} in both numerator and denominator.

$$ab^{3}$$

Expand

$$ab^{3}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -3 and 2 to get -1.

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

To multiply powers of the same base, add their exponents. Add -1 and -1 to get -2.

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

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

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

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

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

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

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

Express \frac{1}{a}b^{2} as a single fraction.

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

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

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

Express \frac{1}{b}a as a single fraction.

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

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

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

Multiply \frac{\left(b^{2}\right)^{4}}{a^{4}} times \frac{a^{2}}{b^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out a^{2} in both numerator and denominator.

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

Express \frac{\left(b^{2}\right)^{4}}{a^{2}b^{2}}a^{3} as a single fraction.

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

Cancel out a^{2} in both numerator and denominator.

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

Express \frac{\frac{a\left(b^{2}\right)^{4}}{b^{2}}}{b^{3}} as a single fraction.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{ab^{8}}{b^{2}b^{3}}$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{ab^{8}}{b^{5}}$$

Cancel out b^{5} in both numerator and denominator.

$$ab^{3}$$