To solve the problem, we need to evaluate the given expression \(\frac{x^{m-n} \times x^m}{x^{2m+n}}\) with the values \(x = 5\), \(m = 6\), and \(n = 2\).
Here's the detailed step-by-step solution:
1. Substitute the given values:
- \(x = 5\)
- \(m = 6\)
- \(n = 2\)
2. Evaluate the numerator:
The numerator is \(x^{m-n} \times x^m\).
- First, we calculate \(m - n\):
[tex]\[
m - n = 6 - 2 = 4
\][/tex]
- Then, we calculate \(x^{m-n}\):
[tex]\[
x^{m-n} = x^4 = 5^4
\][/tex]
Since \(5^4 = 625\), we have:
[tex]\[
x^{m-n} = 625
\][/tex]
- We also need \(x^m\):
[tex]\[
x^m = x^6 = 5^6
\][/tex]
Since \(5^6 = 15625\), we have:
[tex]\[
x^m = 15625
\][/tex]
- Now, we multiply these together:
[tex]\[
x^{m-n} \times x^m = 5^4 \times 5^6
\][/tex]
- Using the properties of exponents (\(x^a \times x^b = x^{a+b}\)), we get:
[tex]\[
5^4 \times 5^6 = 5^{4+6} = 5^{10}
\][/tex]
and \(5^{10} = 9765625\).
3. Evaluate the denominator:
The denominator is \(x^{2m+n}\).
- First, we calculate \(2m + n\):
[tex]\[
2m + n = 2(6) + 2 = 12 + 2 = 14
\][/tex]
- Then, we calculate \(x^{2m+n}\):
[tex]\[
x^{2m+n} = x^{14} = 5^{14}
\][/tex]
Since \(5^{14} = 6103515625\), we have:
[tex]\[
x^{2m+n} = 6103515625
\][/tex]
4. Form the expression with the evaluated values:
The expression now is:
[tex]\[
\frac{5^{10}}{5^{14}} = \frac{9765625}{6103515625}
\][/tex]
5. Simplify the expression:
We can simplify \(\frac{5^{10}}{5^{14}}\) using the properties of exponents (\(\frac{x^a}{x^b} = x^{a-b}\)):
[tex]\[
\frac{5^{10}}{5^{14}} = 5^{10-14} = 5^{-4}
\][/tex]
And \(5^{-4}\) is the same as \(\frac{1}{5^4}\):
[tex]\[
5^{-4} = \frac{1}{5^4} = \frac{1}{625}
\][/tex]
Hence, we get the value:
[tex]\[
\frac{9765625}{6103515625} = \frac{1}{625}
\][/tex]
6. Convert to decimal form:
\(\frac{1}{625} = 0.0016\)
Therefore, the value of the given expression [tex]\(\frac{x^{m-n} \times x^m}{x^{2m+n}}\)[/tex] for [tex]\(x = 5\)[/tex], [tex]\(m = 6\)[/tex], and [tex]\(n = 2\)[/tex] is [tex]\(0.0016\)[/tex].
