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

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

Evaluate

$$\frac{b^{5}}{a^{2}}$$

Solution Steps

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

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

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply 3 and -2 to get -6.

$$\frac{a^{4}b^{-4}c^{-6}}{\left(a^{2}b^{-3}c^{-2}\right)^{3}}$$

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

$$\frac{a^{4}b^{-4}c^{-6}}{\left(a^{2}\right)^{3}\left(b^{-3}\right)^{3}\left(c^{-2}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{a^{4}b^{-4}c^{-6}}{a^{6}\left(b^{-3}\right)^{3}\left(c^{-2}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\frac{a^{4}b^{-4}c^{-6}}{a^{6}b^{-9}\left(c^{-2}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and 3 to get -6.

$$\frac{a^{4}b^{-4}c^{-6}}{a^{6}b^{-9}c^{-6}}$$

Cancel out c^{-6}a^{4} in both numerator and denominator.

$$\frac{b^{-4}}{b^{-9}a^{2}}$$

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

$$\frac{b^{5}}{a^{2}}$$

Expand

$$\frac{b^{5}}{a^{2}}$$

Solution Steps

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

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

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply 3 and -2 to get -6.

$$\frac{a^{4}b^{-4}c^{-6}}{\left(a^{2}b^{-3}c^{-2}\right)^{3}}$$

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

$$\frac{a^{4}b^{-4}c^{-6}}{\left(a^{2}\right)^{3}\left(b^{-3}\right)^{3}\left(c^{-2}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{a^{4}b^{-4}c^{-6}}{a^{6}\left(b^{-3}\right)^{3}\left(c^{-2}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\frac{a^{4}b^{-4}c^{-6}}{a^{6}b^{-9}\left(c^{-2}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and 3 to get -6.

$$\frac{a^{4}b^{-4}c^{-6}}{a^{6}b^{-9}c^{-6}}$$

Cancel out c^{-6}a^{4} in both numerator and denominator.

$$\frac{b^{-4}}{b^{-9}a^{2}}$$

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

$$\frac{b^{5}}{a^{2}}$$