Questions: Simplify. Do not use negative exponents in the answer. 4^-2 b^8 c^5 / 3^-3 b^4 c^7

Solution Steps

To simplify the given expression, we need to handle the exponents separately for each variable and constant. First, convert the negative exponents to positive by taking the reciprocal. Then, simplify the expression by subtracting the exponents of like bases in the numerator and the denominator.

We start with the expression: \[ \frac{4^{-2} b^{8} c^{5}}{3^{-3} b^{4} c^{7}} \] We will convert the negative exponents to positive by taking the reciprocal.

The expression can be rewritten as: \[ \frac{b^{8} c^{5}}{b^{4} c^{7}} \cdot \frac{1}{4^{2}} \cdot 3^{3} \] This simplifies to: \[ \frac{b^{8}}{b^{4}} \cdot \frac{c^{5}}{c^{7}} \cdot \frac{3^{3}}{4^{2}} \]

Now, we simplify the exponents: \[ b^{8 - 4} = b^{4} \quad \text{and} \quad c^{5 - 7} = c^{-2} = \frac{1}{c^{2}} \] Thus, the expression becomes: \[ \frac{3^{3}}{4^{2}} \cdot \frac{b^{4}}{c^{2}} \]

Calculating the numerical coefficient: \[ 3^{3} = 27 \quad \text{and} \quad 4^{2} = 16 \] So, we have: \[ \frac{27}{16} b^{4} c^{-2} = \frac{27 b^{4}}{16 c^{2}} \]

Final Answer