Solve 25/(a^{-2b)^{4}}*(frac{5a^{2}b^-3}{3a^-5b^-2})^-2

$$\frac { 25 } { ( a ^ { - 2 } b ) ^ { 4 } } \times ( \frac { 5 a ^ { 2 } b ^ { - 3 } } { 3 a ^ { - 5 } b ^ { - 2 } } ) ^ { - 2 }$$

Evaluate

$$\frac{9}{b^{2}a^{6}}$$

Solution Steps

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

$$\frac{25}{\left(a^{-2}\right)^{4}b^{4}}\times \left(\frac{5a^{2}b^{-3}}{3a^{-5}b^{-2}}\right)^{-2}$$

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

$$\frac{25}{a^{-8}b^{4}}\times \left(\frac{5a^{2}b^{-3}}{3a^{-5}b^{-2}}\right)^{-2}$$

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

$$\frac{25}{a^{-8}b^{4}}\times \left(\frac{5b^{-3}a^{7}}{3b^{-2}}\right)^{-2}$$

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

$$\frac{25}{a^{-8}b^{4}}\times \left(\frac{5a^{7}}{3b^{1}}\right)^{-2}$$

Calculate b to the power of 1 and get b.

$$\frac{25}{a^{-8}b^{4}}\times \left(\frac{5a^{7}}{3b}\right)^{-2}$$

To raise \frac{5a^{7}}{3b} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{25}{a^{-8}b^{4}}\times \frac{\left(5a^{7}\right)^{-2}}{\left(3b\right)^{-2}}$$

Multiply \frac{25}{a^{-8}b^{4}} times \frac{\left(5a^{7}\right)^{-2}}{\left(3b\right)^{-2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25\times \left(5a^{7}\right)^{-2}}{a^{-8}b^{4}\times \left(3b\right)^{-2}}$$

Expand \left(5a^{7}\right)^{-2}.

$$\frac{25\times 5^{-2}\left(a^{7}\right)^{-2}}{a^{-8}b^{4}\times \left(3b\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -2 to get -14.

$$\frac{25\times 5^{-2}a^{-14}}{a^{-8}b^{4}\times \left(3b\right)^{-2}}$$

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{25\times \frac{1}{25}a^{-14}}{a^{-8}b^{4}\times \left(3b\right)^{-2}}$$

Multiply 25 and \frac{1}{25} to get 1.

$$\frac{a^{-14}}{a^{-8}b^{4}\times \left(3b\right)^{-2}}$$

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

$$\frac{a^{-14}}{a^{-8}b^{4}\times 3^{-2}b^{-2}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{a^{-14}}{a^{-8}b^{4}\times \frac{1}{9}b^{-2}}$$

To multiply powers of the same base, add their exponents. Add 4 and -2 to get 2.

$$\frac{a^{-14}}{a^{-8}b^{2}\times \frac{1}{9}}$$

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

$$\frac{1}{\frac{1}{9}b^{2}a^{6}}$$

Expand

$$\frac{9}{b^{2}a^{6}}$$

Solution Steps

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

$$\frac{25}{\left(a^{-2}\right)^{4}b^{4}}\times \left(\frac{5a^{2}b^{-3}}{3a^{-5}b^{-2}}\right)^{-2}$$

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

$$\frac{25}{a^{-8}b^{4}}\times \left(\frac{5a^{2}b^{-3}}{3a^{-5}b^{-2}}\right)^{-2}$$

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

$$\frac{25}{a^{-8}b^{4}}\times \left(\frac{5b^{-3}a^{7}}{3b^{-2}}\right)^{-2}$$

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

$$\frac{25}{a^{-8}b^{4}}\times \left(\frac{5a^{7}}{3b^{1}}\right)^{-2}$$

Calculate b to the power of 1 and get b.

$$\frac{25}{a^{-8}b^{4}}\times \left(\frac{5a^{7}}{3b}\right)^{-2}$$

To raise \frac{5a^{7}}{3b} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{25}{a^{-8}b^{4}}\times \frac{\left(5a^{7}\right)^{-2}}{\left(3b\right)^{-2}}$$

Multiply \frac{25}{a^{-8}b^{4}} times \frac{\left(5a^{7}\right)^{-2}}{\left(3b\right)^{-2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25\times \left(5a^{7}\right)^{-2}}{a^{-8}b^{4}\times \left(3b\right)^{-2}}$$

Expand \left(5a^{7}\right)^{-2}.

$$\frac{25\times 5^{-2}\left(a^{7}\right)^{-2}}{a^{-8}b^{4}\times \left(3b\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -2 to get -14.

$$\frac{25\times 5^{-2}a^{-14}}{a^{-8}b^{4}\times \left(3b\right)^{-2}}$$

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{25\times \frac{1}{25}a^{-14}}{a^{-8}b^{4}\times \left(3b\right)^{-2}}$$

Multiply 25 and \frac{1}{25} to get 1.

$$\frac{a^{-14}}{a^{-8}b^{4}\times \left(3b\right)^{-2}}$$

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

$$\frac{a^{-14}}{a^{-8}b^{4}\times 3^{-2}b^{-2}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{a^{-14}}{a^{-8}b^{4}\times \frac{1}{9}b^{-2}}$$

To multiply powers of the same base, add their exponents. Add 4 and -2 to get 2.

$$\frac{a^{-14}}{a^{-8}b^{2}\times \frac{1}{9}}$$

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

$$\frac{1}{\frac{1}{9}b^{2}a^{6}}$$