Sure, let's break down the given expression step by step.
We start with the expression:
[tex]\[
\frac{8}{6} \sqrt{\frac{7}{8}} \times \frac{8}{8} \sqrt{\frac{4}{7}}
\][/tex]
1. Simplify the Fractions:
- [tex]\(\frac{8}{6}\)[/tex]:
Simplify by dividing both numerator and denominator by their greatest common divisor (GCD), which is 2.
[tex]\[
\frac{8}{6} = \frac{8 \div 2}{6 \div 2} = \frac{4}{3} \approx 1.3333333333333333
\][/tex]
- [tex]\(\frac{8}{8}\)[/tex]:
The numerator and the denominator are the same, so it simplifies to 1.
[tex]\[
\frac{8}{8} = 1.0
\][/tex]
2. Evaluate the Square Roots:
- [tex]\(\sqrt{\frac{7}{8}}\)[/tex]:
Take the square root of [tex]\(\frac{7}{8}\)[/tex].
[tex]\[
\sqrt{\frac{7}{8}} \approx 0.9354143466934853
\][/tex]
- [tex]\(\sqrt{\frac{4}{7}}\)[/tex]:
Take the square root of [tex]\(\frac{4}{7}\)[/tex].
[tex]\[
\sqrt{\frac{4}{7}} \approx 0.7559289460184544
\][/tex]
3. Calculate the Product:
Now, multiply all the simplified parts together:
[tex]\[
\frac{4}{3} \times 0.9354143466934853 \times 1 \times 0.7559289460184544
\][/tex]
Step-by-step:
- First, multiply [tex]\(\frac{4}{3}\)[/tex] by [tex]\(\sqrt{\frac{7}{8}}\)[/tex]:
[tex]\[
\frac{4}{3} \times 0.9354143466934853 \approx 1.3333333333333333 \times 0.9354143466934853 \approx 1.247218462257981
\][/tex]
- Since [tex]\(\frac{8}{8}\)[/tex] is 1, multiplying by 1 does not change the value.
So:
[tex]\[
1.247218462257981 \times 1 = 1.247218462257981
\][/tex]
- Finally, multiply by [tex]\(\sqrt{\frac{4}{7}}\)[/tex]:
[tex]\[
1.247218462257981 \times 0.7559289460184544 \approx 0.9428090415820632
\][/tex]
Thus, the value of the given expression is approximately:
[tex]\[
0.9428090415820632
\][/tex]
