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Chapter 6: Problem 16

Find each product or quotient. Simplify your answers. $$ \frac{m}{4} \div \frac{4}{m} $$

Short Answer

Expert verified
\( \frac{m^2}{16} \)

Step by step solution

01

Understand the division of fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
02

Find the reciprocal of the second fraction

The second fraction is \(\frac{4}{m}\). The reciprocal is \(\frac{m}{4}\).
03

Set up the multiplication

Rewrite the division problem as a multiplication problem: \(\frac{m}{4} \times \frac{m}{4}\).
04

Multiply the fractions

Multiply the numerators together and the denominators together: \[ \frac{m \times m}{4 \times 4} = \frac{m^2}{16} \]
05

Simplify the expression

The expression \(\frac{m^2}{16}\) is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal of a fraction
The reciprocal of a fraction is simply found by flipping its numerator and denominator. For example, the reciprocal of \( \frac{4}{m} \) is \( \frac{m}{4} \).
This is a key step when dividing fractions, as it turns division into multiplication, which is often easier to handle.
In our problem, we need to divide \( \frac{m}{4} \) by \( \frac{4}{m} \).
To do this, we multiply \( \frac{m}{4} \) by the reciprocal of \( \frac{4}{m} \), which is \( \frac{m}{4} \).
This transforms our original division problem into a simpler multiplication problem.
Multiplication of fractions
When multiplying fractions, you multiply the numerators together and the denominators together. For our problem, we rewrite the operation as \( \frac{m}{4} \times \frac{m}{4} \).
This is how you do it:
\ \begin{align*} \frac{m}{4} \times \frac{m}{4} &= \frac{m \times m}{4 \times 4} \ &= \frac{m^2}{16}. \ \end{align*}
Multiplying two fractions involves just these simple steps. Make sure each fraction is in its simplest form before you multiply them.
Always double-check your work to ensure that your multiplication was correct.
Simplifying expressions
Simplifying an expression means reducing it to its most basic form. In our problem, we end up with the fraction \( \frac{m^2}{16} \).
We need to check if this fraction can be simplified further.
\ \begin{itemize} \item Check for common factors in the numerator and denominator. \item If there are none, then the fraction is already in its simplest form. \end{itemize}
In this case, \( m^2 \) and 16 have no common factors besides 1, so \( \frac{m^2}{16} \) is already simplified.
Keep an eye out for opportunities to simplify at every step to make the final answer easier to understand.

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Most popular questions from this chapter

Rewrite each expression using a single base and a single exponent. $$ a^{12} \cdot\left(a^{2}\right)^{-7} $$

Write each expression as the product of binomials. $$ 16-25 x^{2} $$

Expand and simplify each expression. $$ \left(q-\frac{1}{4}\right)^{2} $$

Expand each expression. Simplify your results by combining like terms. \(2(x+y)+x(3+y)(x+2)\)

Solve each equation. \(3.2-2 b=1.1\)
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