Solve frac{(-16a^{5}x^{3})^{3}}{(4a^{3}x^4)^3(-2ax^3)^5}

$$\frac { ( - 16 a ^ { 5 } x ^ { 3 } ) ^ { 3 } } { ( 4 a ^ { 3 } x ^ { 4 } ) ^ { 3 } ( - 2 a x ^ { 3 } ) ^ { 5 } }$$

Evaluate

$$\frac{2a}{x^{18}}$$

Solution Steps

Expand \left(-16a^{5}x^{3}\right)^{3}.

$$\frac{\left(-16\right)^{3}\left(a^{5}\right)^{3}\left(x^{3}\right)^{3}}{\left(4a^{3}x^{4}\right)^{3}\left(-2ax^{3}\right)^{5}}$$

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

$$\frac{\left(-16\right)^{3}a^{15}\left(x^{3}\right)^{3}}{\left(4a^{3}x^{4}\right)^{3}\left(-2ax^{3}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{\left(-16\right)^{3}a^{15}x^{9}}{\left(4a^{3}x^{4}\right)^{3}\left(-2ax^{3}\right)^{5}}$$

Calculate -16 to the power of 3 and get -4096.

$$\frac{-4096a^{15}x^{9}}{\left(4a^{3}x^{4}\right)^{3}\left(-2ax^{3}\right)^{5}}$$

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

$$\frac{-4096a^{15}x^{9}}{4^{3}\left(a^{3}\right)^{3}\left(x^{4}\right)^{3}\left(-2ax^{3}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{-4096a^{15}x^{9}}{4^{3}a^{9}\left(x^{4}\right)^{3}\left(-2ax^{3}\right)^{5}}$$

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

$$\frac{-4096a^{15}x^{9}}{4^{3}a^{9}x^{12}\left(-2ax^{3}\right)^{5}}$$

Calculate 4 to the power of 3 and get 64.

$$\frac{-4096a^{15}x^{9}}{64a^{9}x^{12}\left(-2ax^{3}\right)^{5}}$$

Expand \left(-2ax^{3}\right)^{5}.

$$\frac{-4096a^{15}x^{9}}{64a^{9}x^{12}\left(-2\right)^{5}a^{5}\left(x^{3}\right)^{5}}$$

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

$$\frac{-4096a^{15}x^{9}}{64a^{9}x^{12}\left(-2\right)^{5}a^{5}x^{15}}$$

Calculate -2 to the power of 5 and get -32.

$$\frac{-4096a^{15}x^{9}}{64a^{9}x^{12}\left(-32\right)a^{5}x^{15}}$$

Multiply 64 and -32 to get -2048.

$$\frac{-4096a^{15}x^{9}}{-2048a^{9}x^{12}a^{5}x^{15}}$$

To multiply powers of the same base, add their exponents. Add 9 and 5 to get 14.

$$\frac{-4096a^{15}x^{9}}{-2048a^{14}x^{12}x^{15}}$$

To multiply powers of the same base, add their exponents. Add 12 and 15 to get 27.

$$\frac{-4096a^{15}x^{9}}{-2048a^{14}x^{27}}$$

Cancel out 2048x^{9}a^{14} in both numerator and denominator.

$$\frac{-2a}{-x^{18}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{2a}{x^{18}}$$

Expand

$$\frac{2a}{x^{18}}$$

Solution Steps

Expand \left(-16a^{5}x^{3}\right)^{3}.

$$\frac{\left(-16\right)^{3}\left(a^{5}\right)^{3}\left(x^{3}\right)^{3}}{\left(4a^{3}x^{4}\right)^{3}\left(-2ax^{3}\right)^{5}}$$

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

$$\frac{\left(-16\right)^{3}a^{15}\left(x^{3}\right)^{3}}{\left(4a^{3}x^{4}\right)^{3}\left(-2ax^{3}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{\left(-16\right)^{3}a^{15}x^{9}}{\left(4a^{3}x^{4}\right)^{3}\left(-2ax^{3}\right)^{5}}$$

Calculate -16 to the power of 3 and get -4096.

$$\frac{-4096a^{15}x^{9}}{\left(4a^{3}x^{4}\right)^{3}\left(-2ax^{3}\right)^{5}}$$

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

$$\frac{-4096a^{15}x^{9}}{4^{3}\left(a^{3}\right)^{3}\left(x^{4}\right)^{3}\left(-2ax^{3}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{-4096a^{15}x^{9}}{4^{3}a^{9}\left(x^{4}\right)^{3}\left(-2ax^{3}\right)^{5}}$$

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

$$\frac{-4096a^{15}x^{9}}{4^{3}a^{9}x^{12}\left(-2ax^{3}\right)^{5}}$$

Calculate 4 to the power of 3 and get 64.

$$\frac{-4096a^{15}x^{9}}{64a^{9}x^{12}\left(-2ax^{3}\right)^{5}}$$

Expand \left(-2ax^{3}\right)^{5}.

$$\frac{-4096a^{15}x^{9}}{64a^{9}x^{12}\left(-2\right)^{5}a^{5}\left(x^{3}\right)^{5}}$$

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

$$\frac{-4096a^{15}x^{9}}{64a^{9}x^{12}\left(-2\right)^{5}a^{5}x^{15}}$$

Calculate -2 to the power of 5 and get -32.

$$\frac{-4096a^{15}x^{9}}{64a^{9}x^{12}\left(-32\right)a^{5}x^{15}}$$

Multiply 64 and -32 to get -2048.

$$\frac{-4096a^{15}x^{9}}{-2048a^{9}x^{12}a^{5}x^{15}}$$

To multiply powers of the same base, add their exponents. Add 9 and 5 to get 14.

$$\frac{-4096a^{15}x^{9}}{-2048a^{14}x^{12}x^{15}}$$

To multiply powers of the same base, add their exponents. Add 12 and 15 to get 27.

$$\frac{-4096a^{15}x^{9}}{-2048a^{14}x^{27}}$$

Cancel out 2048x^{9}a^{14} in both numerator and denominator.

$$\frac{-2a}{-x^{18}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{2a}{x^{18}}$$