Let's break down the steps to solve the given problem using the multiplicative inverse property.
Step 1: Understanding the Fractions
We are given two fractions:
[tex]\[ \frac{6}{13} \quad \text{and} \quad -\frac{7}{16} \][/tex]
Step 2: Finding the Reciprocal
To multiply by the reciprocal of the second fraction, we first need to find the reciprocal of [tex]\( -\frac{7}{16} \)[/tex]. The reciprocal of a fraction [tex]\( \frac{a}{b} \)[/tex] is [tex]\( \frac{b}{a} \)[/tex].
Hence, the reciprocal of [tex]\( -\frac{7}{16} \)[/tex] is:
[tex]\[ -\frac{16}{7} \][/tex]
Step 3: Multiplication of Fractions
Next, we need to multiply the first fraction [tex]\( \frac{6}{13} \)[/tex] by the reciprocal of the second fraction [tex]\( -\frac{16}{7} \)[/tex].
To multiply two fractions, we multiply the numerators together and the denominators together:
[tex]\[
\frac{6}{13} \times -\frac{16}{7} = \frac{6 \times -16}{13 \times 7}
\][/tex]
When we carry out the multiplication:
[tex]\[
6 \times -16 = -96 \quad \text{and} \quad 13 \times 7 = 91
\][/tex]
So, the resulting fraction is:
[tex]\[
\frac{-96}{91}
\][/tex]
Step 4: Simplifying the Result
Although [tex]\(\frac{-96}{91}\)[/tex] is already in its simplest form, for practical purposes, we may also express it as a decimal. Let's approximate the decimal value of this fraction:
[tex]\[
\frac{-96}{91} \approx -1.054945054945055
\][/tex]
So the multiplication of [tex]\( \frac{6}{13} \)[/tex] and the reciprocal of [tex]\( -\frac{7}{16} \)[/tex], which is [tex]\( -\frac{16}{7} \)[/tex], results in:
[tex]\[
\frac{-96}{91} \approx -1.054945054945055
\][/tex]
To summarize, the reciprocal of [tex]\( -\frac{7}{16} \)[/tex] is [tex]\( -\frac{16}{7} \)[/tex]. When we multiply [tex]\( \frac{6}{13} \)[/tex] by this reciprocal, we get approximately [tex]\( -1.054945054945055 \)[/tex].
