Solve (frac{-15a^{4}b^{2}}{5a^{10}b^-3})^3(4a^3)^-2

$$( \frac { - 15 a ^ { 4 } b ^ { 2 } } { 5 a ^ { 10 } b ^ { - 3 } } ) ^ { 3 } ( 4 a ^ { 3 } ) ^ { - 2 }$$

Evaluate

$$-\frac{27b^{15}}{16a^{24}}$$

Solution Steps

Cancel out 5a^{4} in both numerator and denominator.

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

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

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

To raise \frac{-3b^{5}}{a^{6}} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

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

$$\frac{\left(-3b^{5}\right)^{3}}{\left(a^{6}\right)^{3}}\times \frac{1}{16}a^{-6}$$

Multiply \frac{\left(-3b^{5}\right)^{3}}{\left(a^{6}\right)^{3}} times \frac{1}{16} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-3b^{5}\right)^{3}}{\left(a^{6}\right)^{3}\times 16}a^{-6}$$

Express \frac{\left(-3b^{5}\right)^{3}}{\left(a^{6}\right)^{3}\times 16}a^{-6} as a single fraction.

$$\frac{\left(-3b^{5}\right)^{3}a^{-6}}{\left(a^{6}\right)^{3}\times 16}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 3 to get 18.

$$\frac{\left(-3b^{5}\right)^{3}a^{-6}}{a^{18}\times 16}$$

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

$$\frac{\left(-3b^{5}\right)^{3}}{16a^{24}}$$

Expand \left(-3b^{5}\right)^{3}.

$$\frac{\left(-3\right)^{3}\left(b^{5}\right)^{3}}{16a^{24}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{\left(-3\right)^{3}b^{15}}{16a^{24}}$$

Calculate -3 to the power of 3 and get -27.

$$\frac{-27b^{15}}{16a^{24}}$$

Expand

$$-\frac{27b^{15}}{16a^{24}}$$

Solution Steps

Cancel out 5a^{4} in both numerator and denominator.

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

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

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

To raise \frac{-3b^{5}}{a^{6}} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

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

$$\frac{\left(-3b^{5}\right)^{3}}{\left(a^{6}\right)^{3}}\times \frac{1}{16}a^{-6}$$

Multiply \frac{\left(-3b^{5}\right)^{3}}{\left(a^{6}\right)^{3}} times \frac{1}{16} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-3b^{5}\right)^{3}}{\left(a^{6}\right)^{3}\times 16}a^{-6}$$

Express \frac{\left(-3b^{5}\right)^{3}}{\left(a^{6}\right)^{3}\times 16}a^{-6} as a single fraction.

$$\frac{\left(-3b^{5}\right)^{3}a^{-6}}{\left(a^{6}\right)^{3}\times 16}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 3 to get 18.

$$\frac{\left(-3b^{5}\right)^{3}a^{-6}}{a^{18}\times 16}$$

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

$$\frac{\left(-3b^{5}\right)^{3}}{16a^{24}}$$

Expand \left(-3b^{5}\right)^{3}.

$$\frac{\left(-3\right)^{3}\left(b^{5}\right)^{3}}{16a^{24}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\frac{\left(-3\right)^{3}b^{15}}{16a^{24}}$$

Calculate -3 to the power of 3 and get -27.

$$\frac{-27b^{15}}{16a^{24}}$$