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Evaluate the following expression:

[tex]\[ \left(\frac{27}{125}\right)^{\frac{2}{3}} \times \left(\frac{9}{25}\right)^{\frac{-3}{2}} \][/tex]

Sagot :

Let's solve the given expression step by step:

[tex]\[ \left(\frac{27}{125}\right)^{\frac{2}{3}} \times \left(\frac{9}{25}\right)^{\frac{-3}{2}} \][/tex]

### Step 1: Simplify the fractions separately

First, let's evaluate the fraction [tex]\(\frac{27}{125}\)[/tex]:

The fraction [tex]\(\frac{27}{125}\)[/tex] can be simplified to:
[tex]\[ \frac{27}{125} = 0.216 \][/tex]

Next, let's evaluate the fraction [tex]\(\frac{9}{25}\)[/tex]:

The fraction [tex]\(\frac{9}{25}\)[/tex] can be simplified to:
[tex]\[ \frac{9}{25} = 0.36 \][/tex]

### Step 2: Compute the powers of each fraction

#### Power Calculation for [tex]\(\frac{27}{125}\)[/tex]

Now, let's compute [tex]\( \left(\frac{27}{125}\right)^{\frac{2}{3}} \)[/tex]:

[tex]\[ 0.216^{\frac{2}{3}} = 0.36000000000000004 \][/tex]

#### Power Calculation for [tex]\(\frac{9}{25}\)[/tex]

Next, let's compute [tex]\( \left(\frac{9}{25}\right)^{\frac{-3}{2}} \)[/tex]:

[tex]\[ 0.36^{-\frac{3}{2}} = 4.62962962962963 \][/tex]

### Step 3: Multiply the results

Finally, multiply these computed values together:

[tex]\[ 0.36000000000000004 \times 4.62962962962963 = 1.666666666666667 \][/tex]

### Conclusion

Thus, the value of the expression [tex]\(\left(\frac{27}{125}\right)^{\frac{2}{3}} \times \left(\frac{9}{25}\right)^{\frac{-3}{2}}\)[/tex] is:

[tex]\[ 1.666666666666667 \][/tex]