Let's break this problem down step by step, considering the given values \( m = 3 \) and \( n = -2 \).
We need to calculate the value of the expression:
[tex]\[ 125^{\frac{2}{m}} \times 36^{\left(-\frac{1}{n}\right)} \div 16^{\frac{m}{2 n}} \][/tex]
1. Evaluate \( 125^{\frac{2}{m}} \):
[tex]\[ 125^{\frac{2}{m}} = 125^{\frac{2}{3}} \][/tex]
Since \(125 = 5^3\), we can simplify this as:
[tex]\[ 125^{\frac{2}{3}} = (5^3)^{\frac{2}{3}} \][/tex]
Using the property of exponents \( (a^b)^c = a^{bc} \):
[tex]\[ (5^3)^{\frac{2}{3}} = 5^{3 \cdot \frac{2}{3}} = 5^2 = 25 \][/tex]
2. Evaluate \( 36^{\left(-\frac{1}{n}\right)} \):
[tex]\[ 36^{\left(-\frac{1}{n}\right)} = 36^{\left(-\frac{1}{-2}\right)} = 36^{\frac{1}{2}} \][/tex]
The expression \(36^{\frac{1}{2}}\) represents the square root of 36:
[tex]\[ 36^{\frac{1}{2}} = \sqrt{36} = 6 \][/tex]
3. Evaluate \( 16^{\frac{m}{2 n}} \):
[tex]\[ 16^{\frac{m}{2 n}} = 16^{\frac{3}{2 \cdot (-2)}} = 16^{\frac{3}{-4}} = 16^{-\frac{3}{4}} \][/tex]
We know \(16 = 2^4\), so:
[tex]\[ 16^{-\frac{3}{4}} = (2^4)^{-\frac{3}{4}} = 2^{4 \cdot -\frac{3}{4}} = 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \][/tex]
Simplifying further:
[tex]\[ \frac{1}{8} = 0.125 \][/tex]
Now, combine all parts according to the given expression:
[tex]\[ 125^{\frac{2}{3}} \times 36^{\left(\frac{1}{2}\right)} \div 16^{\frac{3}{-4}} = 25 \times 6 \div 0.125 \][/tex]
First, calculate \( 25 \times 6 \):
[tex]\[ 25 \times 6 = 150 \][/tex]
Next, divide this by \( 0.125 \):
[tex]\[ 150 \div 0.125 = 150 \times \frac{1}{0.125} = 150 \times 8 = 1200 \][/tex]
Therefore, the value of the expression is:
[tex]\[ \boxed{1200} \][/tex]
To recap:
- \( 125^{\frac{2}{3}} = 25 \)
- \( 36^{\frac{1}{2}} = 6 \)
- \( 16^{-\frac{3}{4}} = 0.125 \)
The final combined result is:
[tex]\[ 25 \times 6 \div 0.125 = 1200 \][/tex]
