Select the correct answer.
Which expression is equivalent to $\frac{m^3 n^3}{m^4 n^3}$ ? Assume that the denominator does not equal zero.
A. $\frac{1}{m n}$
B. $\frac{1}{m}$
C. $\frac{n}{m}$
D. $\frac{m}{n}
Answer (2)
Rewrite the expression as a product of fractions: m 4 n 3 m 3 n 3 = m 4 m 3 ⋅ n 3 n 3 .
Simplify the fraction with m : m 4 m 3 = m 1 .
Simplify the fraction with n : n 3 n 3 = 1 .
Multiply the simplified fractions: m 1 ⋅ 1 = m 1 . The answer is m 1 .
Explanation
Understanding the Problem We are asked to find an expression equivalent to m 4 n 3 m 3 n 3 , given that the denominator is not zero. This means that m = 0 and n = 0 .
Rewriting the Expression To simplify the expression, we can divide out common factors in the numerator and denominator. We can rewrite the expression as a product of two fractions: m 4 n 3 m 3 n 3 = m 4 m 3 ⋅ n 3 n 3
Simplifying Fractions Now, let's simplify each fraction separately. For the first fraction, we have: m 4 m 3 = m 3 ⋅ m m 3 = m 1 For the second fraction, we have: n 3 n 3 = 1
Final Simplification Now, we multiply the simplified fractions together: m 1 ⋅ 1 = m 1 Therefore, the expression m 4 n 3 m 3 n 3 simplifies to m 1 .
Selecting the Correct Answer The equivalent expression is m 1 , which corresponds to option B.
Examples
Imagine you're organizing a bookshelf. You have m 4 n 3 books, where m represents the number of shelves and n represents the number of books on each shelf. If you decide to remove some books, leaving only m 3 n 3 books, the expression m 4 n 3 m 3 n 3 tells you what fraction of the original number of books you have left. Simplifying this expression to m 1 shows that you now have one book for every m books you started with. This kind of simplification is useful in many real-world scenarios where you want to understand proportions or ratios.
The expression m 4 n 3 m 3 n 3 simplifies to m 1 . Therefore, the correct answer is option B: m 1 .
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