Example Question - simplify exponential expression

Simplifying Exponential Expression

The expression given in the image is: \[ \left( \frac{8y^4z^8}{16y^8z} \right)^4 \] To simplify this expression, first, simplify the fraction by canceling common factors and then apply the exponent of 4: \[ \left( \frac{8}{16} \cdot \frac{y^4}{y^8} \cdot \frac{z^8}{z} \right)^4 \] Simplify the fractions: \[ \left( \frac{1}{2} \cdot y^{4-8} \cdot z^{8-1} \right)^4 \] which simplifies further to: \[ \left( \frac{1}{2} \cdot y^{-4} \cdot z^7 \right)^4 \] Now apply the exponent of 4 to each term within the parentheses: \[ \left( \frac{1}{2} \right)^4 \cdot y^{-4 \cdot 4} \cdot z^{7 \cdot 4} \] This gives: \[ \frac{1}{16} \cdot y^{-16} \cdot z^{28} \] Since y has a negative exponent, it can be moved to the denominator: \[ \frac{z^{28}}{16y^{16}} \] This is the simplified form of the original expression.