Question - Simplifying Exponential Expression
Solution:
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.
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