Certainly! Let's break down the given expression step-by-step:
The expression we need to evaluate is:
[tex]$\left[8^{-\frac{2}{3}} \times 2^{\frac{1}{2}} \times 25^{-\frac{5}{4}}\right] \div\left[32^{-\frac{2}{5}} \times 125^{-\frac{5}{6}}\right].$[/tex]
### Step-by-Step Solution:
#### 1. Simplify the numerator.
The numerator can be expressed as:
[tex]$8^{-\frac{2}{3}} \times 2^{\frac{1}{2}} \times 25^{-\frac{5}{4}}.$[/tex]
Let's simplify each term separately.
- [tex]\( 8 = 2^3 \)[/tex], so:
[tex]\[
8^{-\frac{2}{3}} = (2^3)^{-\frac{2}{3}} = 2^{3 \times -\frac{2}{3}} = 2^{-2} = \frac{1}{4}.
\][/tex]
- [tex]\( 2^{\frac{1}{2}} \)[/tex] remains as it is, which is [tex]\( \sqrt{2} \)[/tex].
- [tex]\( 25 = 5^2 \)[/tex], so:
[tex]\[
25^{-\frac{5}{4}} = (5^2)^{-\frac{5}{4}} = 5^{2 \times -\frac{5}{4}} = 5^{-\frac{5}{2}}.
\][/tex]
Thus, the simplified numerator is:
[tex]\[
\frac{1}{4} \times \sqrt{2} \times 5^{-\frac{5}{2}}.
\][/tex]
#### 2. Simplify the denominator.
The denominator can be expressed as:
[tex]$32^{-\frac{2}{5}} \times 125^{-\frac{5}{6}}.$[/tex]
Let's simplify each term separately.
- [tex]\( 32 = 2^5 \)[/tex], so:
[tex]\[
32^{-\frac{2}{5}} = (2^5)^{-\frac{2}{5}} = 2^{5 \times -\frac{2}{5}} = 2^{-2} = \frac{1}{4}.
\][/tex]
- [tex]\( 125 = 5^3 \)[/tex], so:
[tex]\[
125^{-\frac{5}{6}} = (5^3)^{-\frac{5}{6}} = 5^{3 \times -\frac{5}{6}} = 5^{-\frac{5}{2}}.
\][/tex]
Thus, the simplified denominator is:
[tex]\[
\frac{1}{4} \times 5^{-\frac{5}{2}}.
\][/tex]
#### 3. Combine the simplified terms.
Now, rewrite the original expression using the simplified numerator and denominator:
[tex]\[
\frac{\frac{1}{4} \times \sqrt{2} \times 5^{-\frac{5}{2}}}{\frac{1}{4} \times 5^{-\frac{5}{2}}}.
\][/tex]
#### 4. Simplify the fraction.
Notice that both the numerator and the denominator have the common factors [tex]\(\frac{1}{4}\)[/tex] and [tex]\(5^{-\frac{5}{2}}\)[/tex]. So, these common factors will cancel out:
[tex]\[
\frac{\left(\frac{1}{4} \times \sqrt{2} \times 5^{-\frac{5}{2}}\right)}{\left(\frac{1}{4} \times 5^{-\frac{5}{2}}\right)} = \sqrt{2}.
\][/tex]
Therefore, the simplified value of the given expression is:
[tex]\[
\sqrt{2}.
\][/tex]
So, the final result is [tex]\(\boxed{\sqrt{2}}\)[/tex].
