Question

Simplify the expression. The simplified expression should have no negative exponents. $$ \frac{16 x^{5} y^{-8}}{x^{7} y^{4}} \cdot\left(\frac{x^{3} y^{2}}{8 x y}\right)^{4} $$

Short answer

The simplified expression is \( \frac{x^6}{256y^8} \).

Step-by-step solution

Simplify the first fraction

To simplify the first fraction, subtract the exponents of like terms from the numerator and denominator. Hence, \( \frac{16 x^{5} y^{-8}}{x^{7} y^{4}} \) simplifies to \( 16x^{-2}y^{-12} \) or \( \frac{16}{x^2 y^{12}} \).

Simplify the second expression

Raise the entire inner expression to the power of 4, multiplying all exponents inside the brackets by 4. Hence, \( \left( \frac{x^3 y^2}{8x y} \right)^4 \) is equal to \( \frac{x^{12} y^{8}}{4096x^4 y^4} \). Reduce to \( \frac{x^8 y^4}{4096} \).

Multiplication of the parts obtained

Multiply the expressions obtained in step 1 and step 2: \( \frac{16}{x^2 y^{12}} \cdot \frac{x^8 y^4}{4096} = \frac{16x^8 y^4}{4096x^2 y^{12}} \). Simplify by subtracting the exponents to get \( \frac{x^6}{256y^8} \).