Simplify $\frac{3 m^{-2}}{5 n^{-3}}$.
A) $\frac{3 n^3}{5 m^2}$
B) $\frac{1}{m^3 n^4}$
C) $\frac{3 m^2 n^{10}}{2}$
D) $\frac{-4}{5 n^8}$
Answer (2)
To simplify the expression 5 n − 3 3 m − 2 :
Rewrite negative exponents using the property a − n = a n 1 .
Substitute the rewritten terms into the expression.
Divide the fractions by multiplying by the reciprocal.
Simplify the expression to obtain the final result: 5 m 2 3 n 3 .
Explanation
Understanding the Problem We are given the expression 5 n − 3 3 m − 2 and asked to simplify it. We need to simplify the expression by using the property a − n = a n 1 .
Rewriting with Positive Exponents We will use the property a − n = a n 1 to rewrite m − 2 as m 2 1 and n − 3 as n 3 1 .
Substitution Substitute these into the expression to get 5 ( n 3 1 ) 3 ( m 2 1 ) .
Simplifying the Expression Simplify the expression to n 3 5 m 2 3 .
Dividing Fractions Divide the fractions by multiplying by the reciprocal: m 2 3 ⋅ 5 n 3 .
Final Simplification Simplify to get 5 m 2 3 n 3 . Therefore, the simplified expression is 5 m 2 3 n 3 . The correct answer is A.
Examples
Understanding how to simplify expressions with negative exponents is useful in various scientific and engineering fields. For example, in physics, you might encounter expressions with negative exponents when dealing with units or formulas involving inverse relationships, such as the inverse square law for gravitational force. Simplifying these expressions allows for easier calculations and a better understanding of the relationships between different variables. This skill is also crucial in computer science when working with memory addresses or data storage sizes.
The expression 5 n − 3 3 m − 2 simplifies to 5 m 2 3 n 3 , which is option A. By applying the property of negative exponents, we rewrote and simplified the fraction step by step. This process enabled us to express the original fraction in terms of positive exponents.
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