Solve frac{(a^{3}b^{2}c)^{-1}}{(a^{-2}b^{4}c)^{-3}}*(frac{a^{2}b^-3}{a^-4b^5})^-2

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

Evaluate

$$\frac{c^{2}b^{26}}{a^{21}}$$

Solution Steps

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

$$\frac{\left(a^{3}\right)^{-1}\left(b^{2}\right)^{-1}c^{-1}}{\left(a^{-2}b^{4}c\right)^{-3}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{a^{-3}\left(b^{2}\right)^{-1}c^{-1}}{\left(a^{-2}b^{4}c\right)^{-3}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{a^{-3}b^{-2}c^{-1}}{\left(a^{-2}b^{4}c\right)^{-3}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{a^{-3}b^{-2}c^{-1}}{\left(a^{-2}\right)^{-3}\left(b^{4}\right)^{-3}c^{-3}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{a^{-3}b^{-2}c^{-1}}{a^{6}\left(b^{4}\right)^{-3}c^{-3}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{a^{-3}b^{-2}c^{-1}}{a^{6}b^{-12}c^{-3}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{a^{-3}c^{2}b^{10}}{a^{6}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{c^{2}b^{10}}{a^{9}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{c^{2}b^{10}}{a^{9}}\times \left(\frac{b^{-3}a^{6}}{b^{5}}\right)^{-2}$$

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

$$\frac{c^{2}b^{10}}{a^{9}}\times \left(\frac{a^{6}}{b^{8}}\right)^{-2}$$

To raise \frac{a^{6}}{b^{8}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{c^{2}b^{10}}{a^{9}}\times \frac{\left(a^{6}\right)^{-2}}{\left(b^{8}\right)^{-2}}$$

Multiply \frac{c^{2}b^{10}}{a^{9}} times \frac{\left(a^{6}\right)^{-2}}{\left(b^{8}\right)^{-2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{c^{2}b^{10}\left(a^{6}\right)^{-2}}{a^{9}\left(b^{8}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -2 to get -12.

$$\frac{c^{2}b^{10}a^{-12}}{a^{9}\left(b^{8}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and -2 to get -16.

$$\frac{c^{2}b^{10}a^{-12}}{a^{9}b^{-16}}$$

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

$$\frac{a^{-12}c^{2}b^{26}}{a^{9}}$$

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

$$\frac{c^{2}b^{26}}{a^{21}}$$

Expand

$$\frac{c^{2}b^{26}}{a^{21}}$$

Solution Steps

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

$$\frac{\left(a^{3}\right)^{-1}\left(b^{2}\right)^{-1}c^{-1}}{\left(a^{-2}b^{4}c\right)^{-3}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{a^{-3}\left(b^{2}\right)^{-1}c^{-1}}{\left(a^{-2}b^{4}c\right)^{-3}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{a^{-3}b^{-2}c^{-1}}{\left(a^{-2}b^{4}c\right)^{-3}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{a^{-3}b^{-2}c^{-1}}{\left(a^{-2}\right)^{-3}\left(b^{4}\right)^{-3}c^{-3}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{a^{-3}b^{-2}c^{-1}}{a^{6}\left(b^{4}\right)^{-3}c^{-3}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{a^{-3}b^{-2}c^{-1}}{a^{6}b^{-12}c^{-3}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{a^{-3}c^{2}b^{10}}{a^{6}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{c^{2}b^{10}}{a^{9}}\times \left(\frac{a^{2}b^{-3}}{a^{-4}b^{5}}\right)^{-2}$$

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

$$\frac{c^{2}b^{10}}{a^{9}}\times \left(\frac{b^{-3}a^{6}}{b^{5}}\right)^{-2}$$

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

$$\frac{c^{2}b^{10}}{a^{9}}\times \left(\frac{a^{6}}{b^{8}}\right)^{-2}$$

To raise \frac{a^{6}}{b^{8}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{c^{2}b^{10}}{a^{9}}\times \frac{\left(a^{6}\right)^{-2}}{\left(b^{8}\right)^{-2}}$$

Multiply \frac{c^{2}b^{10}}{a^{9}} times \frac{\left(a^{6}\right)^{-2}}{\left(b^{8}\right)^{-2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{c^{2}b^{10}\left(a^{6}\right)^{-2}}{a^{9}\left(b^{8}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -2 to get -12.

$$\frac{c^{2}b^{10}a^{-12}}{a^{9}\left(b^{8}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and -2 to get -16.

$$\frac{c^{2}b^{10}a^{-12}}{a^{9}b^{-16}}$$

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

$$\frac{a^{-12}c^{2}b^{26}}{a^{9}}$$

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

$$\frac{c^{2}b^{26}}{a^{21}}$$