To solve this problem, let’s work through it step by step.
First, we need to find the product of [tex]\(\frac{6}{7}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex]:
1. Multiply the two fractions:
[tex]\[
\frac{6}{7} \times \frac{2}{3} = \frac{6 \times 2}{7 \times 3} = \frac{12}{21}
\][/tex]
2. Simplify the fraction:
[tex]\[
\frac{12}{21} = \frac{4}{7} \quad \text{(since both the numerator and the denominator can be divided by 3)}
\][/tex]
So, the product of [tex]\(\frac{6}{7}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex] is [tex]\(\frac{4}{7}\)[/tex].
Next, we need to find the reciprocal of this product. The reciprocal of a number is given by exchanging the numerator and the denominator.
3. Find the reciprocal:
[tex]\[
\text{Reciprocal of } \frac{4}{7} = \frac{7}{4}
\][/tex]
Thus, the reciprocal of [tex]\(\frac{6}{7} \times \frac{2}{3}\)[/tex] is:
[tex]\[
\frac{7}{4}
\][/tex]
Given the answer choices:
- [tex]$\frac{-4}{7}$[/tex]
- [tex]$\frac{-7}{4}$[/tex]
- [tex]$\frac{7}{4}$[/tex]
The correct answer is:
[tex]\[
\boxed{\frac{7}{4}}
\][/tex]
