Solve frac{(-4a^{2}b^{5})^5}{(4a^-3b^7)^-2}

$$\frac { ( - 4 a ^ { 2 } b ^ { 5 } ) ^ { 5 } } { ( 4 a ^ { - 3 } b ^ { 7 } ) ^ { - 2 } }$$

Evaluate

$$-16384a^{4}b^{39}$$

Solution Steps

Expand \left(-4a^{2}b^{5}\right)^{5}.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{\left(-4\right)^{5}a^{10}\left(b^{5}\right)^{5}}{\left(4a^{-3}b^{7}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 5 to get 25.

$$\frac{\left(-4\right)^{5}a^{10}b^{25}}{\left(4a^{-3}b^{7}\right)^{-2}}$$

Calculate -4 to the power of 5 and get -1024.

$$\frac{-1024a^{10}b^{25}}{\left(4a^{-3}b^{7}\right)^{-2}}$$

Expand \left(4a^{-3}b^{7}\right)^{-2}.

$$\frac{-1024a^{10}b^{25}}{4^{-2}\left(a^{-3}\right)^{-2}\left(b^{7}\right)^{-2}}$$

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

$$\frac{-1024a^{10}b^{25}}{4^{-2}a^{6}\left(b^{7}\right)^{-2}}$$

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

$$\frac{-1024a^{10}b^{25}}{4^{-2}a^{6}b^{-14}}$$

Calculate 4 to the power of -2 and get \frac{1}{16}.

$$\frac{-1024a^{10}b^{25}}{\frac{1}{16}a^{6}b^{-14}}$$

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

$$\frac{-1024a^{4}b^{25}}{\frac{1}{16}b^{-14}}$$

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

$$\frac{-1024a^{4}b^{39}}{\frac{1}{16}}$$

Divide -1024a^{4}b^{39} by \frac{1}{16} by multiplying -1024a^{4}b^{39} by the reciprocal of \frac{1}{16}.

$$-1024a^{4}b^{39}\times 16$$

Multiply -1024 and 16 to get -16384.

$$-16384a^{4}b^{39}$$

Expand

$$-16384a^{4}b^{39}$$

Solution Steps

Expand \left(-4a^{2}b^{5}\right)^{5}.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{\left(-4\right)^{5}a^{10}\left(b^{5}\right)^{5}}{\left(4a^{-3}b^{7}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 5 to get 25.

$$\frac{\left(-4\right)^{5}a^{10}b^{25}}{\left(4a^{-3}b^{7}\right)^{-2}}$$

Calculate -4 to the power of 5 and get -1024.

$$\frac{-1024a^{10}b^{25}}{\left(4a^{-3}b^{7}\right)^{-2}}$$

Expand \left(4a^{-3}b^{7}\right)^{-2}.

$$\frac{-1024a^{10}b^{25}}{4^{-2}\left(a^{-3}\right)^{-2}\left(b^{7}\right)^{-2}}$$

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

$$\frac{-1024a^{10}b^{25}}{4^{-2}a^{6}\left(b^{7}\right)^{-2}}$$

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

$$\frac{-1024a^{10}b^{25}}{4^{-2}a^{6}b^{-14}}$$

Calculate 4 to the power of -2 and get \frac{1}{16}.

$$\frac{-1024a^{10}b^{25}}{\frac{1}{16}a^{6}b^{-14}}$$

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

$$\frac{-1024a^{4}b^{25}}{\frac{1}{16}b^{-14}}$$

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

$$\frac{-1024a^{4}b^{39}}{\frac{1}{16}}$$

Divide -1024a^{4}b^{39} by \frac{1}{16} by multiplying -1024a^{4}b^{39} by the reciprocal of \frac{1}{16}.

$$-1024a^{4}b^{39}\times 16$$

Multiply -1024 and 16 to get -16384.

$$-16384a^{4}b^{39}$$