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

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

Evaluate

$$\left(\frac{c}{a}\right)^{10}$$

Solution Steps

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

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

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

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

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

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

Expand \left(a^{4}b^{-2}c^{-2}\right)^{3}.

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

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{a^{2}b^{-6}c^{4}}{a^{12}\left(b^{-2}\right)^{3}\left(c^{-2}\right)^{3}}$$

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

$$\frac{a^{2}b^{-6}c^{4}}{a^{12}b^{-6}\left(c^{-2}\right)^{3}}$$

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

$$\frac{a^{2}b^{-6}c^{4}}{a^{12}b^{-6}c^{-6}}$$

Cancel out b^{-6}a^{2} in both numerator and denominator.

$$\frac{c^{4}}{c^{-6}a^{10}}$$

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

$$\frac{c^{10}}{a^{10}}$$

Expand

$$\left(\frac{c}{a}\right)^{10}$$

Solution Steps

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

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

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

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

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

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

Expand \left(a^{4}b^{-2}c^{-2}\right)^{3}.

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

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{a^{2}b^{-6}c^{4}}{a^{12}\left(b^{-2}\right)^{3}\left(c^{-2}\right)^{3}}$$

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

$$\frac{a^{2}b^{-6}c^{4}}{a^{12}b^{-6}\left(c^{-2}\right)^{3}}$$

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

$$\frac{a^{2}b^{-6}c^{4}}{a^{12}b^{-6}c^{-6}}$$

Cancel out b^{-6}a^{2} in both numerator and denominator.

$$\frac{c^{4}}{c^{-6}a^{10}}$$

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

$$\frac{c^{10}}{a^{10}}$$