Simplify the following expression.
$\begin{array}{c}
\frac{b^{-3} c^0 d^2}{e^{-4}} \\
\frac{d^{[?]} e^{\square}}{b}
\end{array}$
Answer (1)
Simplify c 0 to 1.
Combine the fractions using multiplication.
Apply exponent rules to simplify the expression.
The simplified expression is b − 4 d 2 + x e y + 4 , where x and y are the unknown exponents. b − 4 d 2 + x e y + 4
Explanation
Initial Analysis We are asked to simplify the expression e − 4 b − 3 c 0 d 2 ⋅ b d [ ?] e □ . Let's denote the unknown exponent of d as x and the unknown exponent of e as y . Then the expression becomes e − 4 b − 3 c 0 d 2 ⋅ b d x e y .
Simplifying c 0 First, we simplify c 0 , which is equal to 1. So the expression becomes e − 4 b − 3 ( 1 ) d 2 ⋅ b d x e y = e − 4 b − 3 d 2 ⋅ b d x e y .
Combining Fractions Next, we combine the two fractions into a single fraction: b e − 4 b − 3 d 2 d x e y .
Applying Exponent Rules Now, we use the exponent rules to simplify the expression. Recall that a m ⋅ a n = a m + n and a n a m = a m − n . Thus, we have b − 3 − 1 d 2 + x e y − ( − 4 ) = b − 4 d 2 + x e y + 4 .
Final Simplified Expression The simplified expression is b − 4 d 2 + x e y + 4 . Without knowing the values of x and y , we cannot simplify further. Therefore, the final answer is b − 4 d 2 + x e y + 4 .
Examples
In physics, when dealing with forces and distances, you often encounter expressions with exponents. Simplifying these expressions, like the one we just did, helps in calculating the net force or total energy in a system. For example, if b represents a distance, d represents a force, and e represents energy, the simplified expression could represent a complex physical relationship that becomes easier to analyze after simplification.
