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

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

Evaluate

$$\frac{1}{100000x^{29}a^{30}}$$

Solution Steps

Expand \left(2a^{3}x^{4}\right)^{-5}.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 4 and -5 to get -20.

$$\frac{2^{-5}a^{-15}x^{-20}\times \left(5a^{3}x\right)^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

Calculate 2 to the power of -5 and get \frac{1}{32}.

$$\frac{\frac{1}{32}a^{-15}x^{-20}\times \left(5a^{3}x\right)^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

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

$$\frac{\frac{1}{32}a^{-15}x^{-20}\times 5^{-5}\left(a^{3}\right)^{-5}x^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

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

$$\frac{\frac{1}{32}a^{-15}x^{-20}\times 5^{-5}a^{-15}x^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

Calculate 5 to the power of -5 and get \frac{1}{3125}.

$$\frac{\frac{1}{32}a^{-15}x^{-20}\times \frac{1}{3125}a^{-15}x^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

Multiply \frac{1}{32} and \frac{1}{3125} to get \frac{1}{100000}.

$$\frac{\frac{1}{100000}a^{-15}x^{-20}a^{-15}x^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

To multiply powers of the same base, add their exponents. Add -15 and -15 to get -30.

$$\frac{\frac{1}{100000}a^{-30}x^{-20}x^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

To multiply powers of the same base, add their exponents. Add -20 and -5 to get -25.

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

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

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{\left(a^{4}\right)^{2}\left(x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

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

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{a^{8}\left(x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

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

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{a^{8}x^{8}\left(a^{2}x\right)^{-4}}$$

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

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{a^{8}x^{8}\left(a^{2}\right)^{-4}x^{-4}}$$

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

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{a^{8}x^{8}a^{-8}x^{-4}}$$

Multiply a^{8} and a^{-8} to get 1.

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{x^{8}x^{-4}}$$

To multiply powers of the same base, add their exponents. Add 8 and -4 to get 4.

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{x^{4}}$$

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

$$\frac{\frac{1}{100000}a^{-30}}{x^{29}}$$

Expand

$$\frac{1}{100000x^{29}a^{30}}$$

Solution Steps

Expand \left(2a^{3}x^{4}\right)^{-5}.

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

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

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

To raise a power to another power, multiply the exponents. Multiply 4 and -5 to get -20.

$$\frac{2^{-5}a^{-15}x^{-20}\times \left(5a^{3}x\right)^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

Calculate 2 to the power of -5 and get \frac{1}{32}.

$$\frac{\frac{1}{32}a^{-15}x^{-20}\times \left(5a^{3}x\right)^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

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

$$\frac{\frac{1}{32}a^{-15}x^{-20}\times 5^{-5}\left(a^{3}\right)^{-5}x^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

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

$$\frac{\frac{1}{32}a^{-15}x^{-20}\times 5^{-5}a^{-15}x^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

Calculate 5 to the power of -5 and get \frac{1}{3125}.

$$\frac{\frac{1}{32}a^{-15}x^{-20}\times \frac{1}{3125}a^{-15}x^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

Multiply \frac{1}{32} and \frac{1}{3125} to get \frac{1}{100000}.

$$\frac{\frac{1}{100000}a^{-15}x^{-20}a^{-15}x^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

To multiply powers of the same base, add their exponents. Add -15 and -15 to get -30.

$$\frac{\frac{1}{100000}a^{-30}x^{-20}x^{-5}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

To multiply powers of the same base, add their exponents. Add -20 and -5 to get -25.

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{\left(a^{4}x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

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

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{\left(a^{4}\right)^{2}\left(x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

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

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{a^{8}\left(x^{4}\right)^{2}\left(a^{2}x\right)^{-4}}$$

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

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{a^{8}x^{8}\left(a^{2}x\right)^{-4}}$$

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

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{a^{8}x^{8}\left(a^{2}\right)^{-4}x^{-4}}$$

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

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{a^{8}x^{8}a^{-8}x^{-4}}$$

Multiply a^{8} and a^{-8} to get 1.

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{x^{8}x^{-4}}$$

To multiply powers of the same base, add their exponents. Add 8 and -4 to get 4.

$$\frac{\frac{1}{100000}a^{-30}x^{-25}}{x^{4}}$$

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

$$\frac{\frac{1}{100000}a^{-30}}{x^{29}}$$