Answer :
To find the value of:
[tex]\[ \left(\frac{27}{125}\right)^{\frac{2}{3}} \times\left(\frac{9}{25}\right)^{-\frac{3}{2}}, \][/tex]
we'll need to evaluate each expression separately and then multiply the results.
### Step 1: Evaluate [tex]\(\left(\frac{27}{125}\right)^{\frac{2}{3}}\)[/tex]
The base is [tex]\(\frac{27}{125}\)[/tex] and the exponent is [tex]\(\frac{2}{3}\)[/tex].
1.1. First, express 27 and 125 as powers of their prime factors:
[tex]\[ 27 = 3^3 \quad \text{and} \quad 125 = 5^3 \][/tex]
1.2. Substitute these into the fraction:
[tex]\[ \frac{27}{125} = \frac{3^3}{5^3} = \left(\frac{3}{5}\right)^3 \][/tex]
1.3. Apply the exponent [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ \left(\frac{3}{5}\right)^3\right)^{\frac{2}{3}} = \left(\frac{3}{5}\right)^{3 \times \frac{2}{3}} = \left(\frac{3}{5}\right)^2 \][/tex]
1.4. Simplify the expression:
[tex]\[ \left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25} \][/tex]
So,
[tex]\[ \left(\frac{27}{125}\right)^{\frac{2}{3}} = \frac{9}{25} \][/tex]
### Step 2: Evaluate [tex]\(\left(\frac{9}{25}\right)^{-\frac{3}{2}}\)[/tex]
The base is [tex]\(\frac{9}{25}\)[/tex] and the exponent is [tex]\(-\frac{3}{2}\)[/tex].
2.1. First, express 9 and 25 as powers of their prime factors:
[tex]\[ 9 = 3^2 \quad \text{and} \quad 25 = 5^2 \][/tex]
2.2. Substitute these into the fraction:
[tex]\[ \frac{9}{25} = \frac{3^2}{5^2} = \left(\frac{3}{5}\right)^2 \][/tex]
2.3. Apply the exponent [tex]\(-\frac{3}{2}\)[/tex]:
[tex]\[ \left(\frac{3}{5}\right)^2\right)^{-\frac{3}{2}} = \left(\frac{3}{5}\right)^{2 \times -\frac{3}{2}} = \left(\frac{3}{5}\right)^{-3} \][/tex]
2.4. Simplify by taking the reciprocal and applying the exponent:
[tex]\[ \left(\frac{3}{5}\right)^{-3} = \left(\frac{5}{3}\right)^3 = \frac{5^3}{3^3} = \frac{125}{27} \][/tex]
So,
[tex]\[ \left(\frac{9}{25}\right)^{-\frac{3}{2}} = \frac{125}{27} \][/tex]
### Step 3: Multiply the results from Step 1 and Step 2
[tex]\[ \left(\frac{27}{125}\right)^{\frac{2}{3}} \times \left(\frac{9}{25}\right)^{-\frac{3}{2}} = \frac{9}{25} \times \frac{125}{27} \][/tex]
3.1. Simplify the multiplication:
[tex]\[ \frac{9}{25} \times \frac{125}{27} = \frac{9 \times 125}{25 \times 27} = \frac{1125}{675} \][/tex]
3.2. Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 75:
[tex]\[ \frac{1125 \div 75}{675 \div 75} = \frac{15}{9} = \frac{5}{3} \approx 1.666666666667 \][/tex]
### Final Result:
[tex]\[ \left(\frac{27}{125}\right)^{\frac{2}{3}} \times \left(\frac{9}{25}\right)^{-\frac{3}{2}} = 1.66667 \][/tex]
Thus, the final value is [tex]\(1.66667\)[/tex].
[tex]\[ \left(\frac{27}{125}\right)^{\frac{2}{3}} \times\left(\frac{9}{25}\right)^{-\frac{3}{2}}, \][/tex]
we'll need to evaluate each expression separately and then multiply the results.
### Step 1: Evaluate [tex]\(\left(\frac{27}{125}\right)^{\frac{2}{3}}\)[/tex]
The base is [tex]\(\frac{27}{125}\)[/tex] and the exponent is [tex]\(\frac{2}{3}\)[/tex].
1.1. First, express 27 and 125 as powers of their prime factors:
[tex]\[ 27 = 3^3 \quad \text{and} \quad 125 = 5^3 \][/tex]
1.2. Substitute these into the fraction:
[tex]\[ \frac{27}{125} = \frac{3^3}{5^3} = \left(\frac{3}{5}\right)^3 \][/tex]
1.3. Apply the exponent [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ \left(\frac{3}{5}\right)^3\right)^{\frac{2}{3}} = \left(\frac{3}{5}\right)^{3 \times \frac{2}{3}} = \left(\frac{3}{5}\right)^2 \][/tex]
1.4. Simplify the expression:
[tex]\[ \left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25} \][/tex]
So,
[tex]\[ \left(\frac{27}{125}\right)^{\frac{2}{3}} = \frac{9}{25} \][/tex]
### Step 2: Evaluate [tex]\(\left(\frac{9}{25}\right)^{-\frac{3}{2}}\)[/tex]
The base is [tex]\(\frac{9}{25}\)[/tex] and the exponent is [tex]\(-\frac{3}{2}\)[/tex].
2.1. First, express 9 and 25 as powers of their prime factors:
[tex]\[ 9 = 3^2 \quad \text{and} \quad 25 = 5^2 \][/tex]
2.2. Substitute these into the fraction:
[tex]\[ \frac{9}{25} = \frac{3^2}{5^2} = \left(\frac{3}{5}\right)^2 \][/tex]
2.3. Apply the exponent [tex]\(-\frac{3}{2}\)[/tex]:
[tex]\[ \left(\frac{3}{5}\right)^2\right)^{-\frac{3}{2}} = \left(\frac{3}{5}\right)^{2 \times -\frac{3}{2}} = \left(\frac{3}{5}\right)^{-3} \][/tex]
2.4. Simplify by taking the reciprocal and applying the exponent:
[tex]\[ \left(\frac{3}{5}\right)^{-3} = \left(\frac{5}{3}\right)^3 = \frac{5^3}{3^3} = \frac{125}{27} \][/tex]
So,
[tex]\[ \left(\frac{9}{25}\right)^{-\frac{3}{2}} = \frac{125}{27} \][/tex]
### Step 3: Multiply the results from Step 1 and Step 2
[tex]\[ \left(\frac{27}{125}\right)^{\frac{2}{3}} \times \left(\frac{9}{25}\right)^{-\frac{3}{2}} = \frac{9}{25} \times \frac{125}{27} \][/tex]
3.1. Simplify the multiplication:
[tex]\[ \frac{9}{25} \times \frac{125}{27} = \frac{9 \times 125}{25 \times 27} = \frac{1125}{675} \][/tex]
3.2. Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 75:
[tex]\[ \frac{1125 \div 75}{675 \div 75} = \frac{15}{9} = \frac{5}{3} \approx 1.666666666667 \][/tex]
### Final Result:
[tex]\[ \left(\frac{27}{125}\right)^{\frac{2}{3}} \times \left(\frac{9}{25}\right)^{-\frac{3}{2}} = 1.66667 \][/tex]
Thus, the final value is [tex]\(1.66667\)[/tex].
