Solve frac{ab^{3}c^-2}{(a^frac{2{3}}b^-1c)^3}

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

Evaluate

$$\frac{b^{6}}{ac^{5}}$$

Solution Steps

Expand \left(a^{\frac{2}{3}}b^{-1}c\right)^{3}.

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

To raise a power to another power, multiply the exponents. Multiply \frac{2}{3} and 3 to get 2.

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

To raise a power to another power, multiply the exponents. Multiply -1 and 3 to get -3.

$$\frac{ab^{3}c^{-2}}{a^{2}b^{-3}c^{3}}$$

Cancel out a in both numerator and denominator.

$$\frac{c^{-2}b^{3}}{b^{-3}ac^{3}}$$

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

$$\frac{c^{-2}b^{6}}{ac^{3}}$$

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

$$\frac{b^{6}}{ac^{5}}$$

Expand

$$\frac{b^{6}}{ac^{5}}$$

Solution Steps

Expand \left(a^{\frac{2}{3}}b^{-1}c\right)^{3}.

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

To raise a power to another power, multiply the exponents. Multiply \frac{2}{3} and 3 to get 2.

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

To raise a power to another power, multiply the exponents. Multiply -1 and 3 to get -3.

$$\frac{ab^{3}c^{-2}}{a^{2}b^{-3}c^{3}}$$

Cancel out a in both numerator and denominator.

$$\frac{c^{-2}b^{3}}{b^{-3}ac^{3}}$$

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

$$\frac{c^{-2}b^{6}}{ac^{3}}$$

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

$$\frac{b^{6}}{ac^{5}}$$