To solve the equation [tex]\( x^{\frac{k-m}{2 m}} \times x^{\frac{m-n}{m n}} \times x^{\frac{n-1}{n l}}=1 \)[/tex], we need to simplify the powers of [tex]\(x\)[/tex] and determine the conditions that make the equation true. Here’s a step-by-step approach:
1. Combine the Exponents:
Recall that when you multiply exponentiated terms with the same base, you add the exponents:
[tex]\[ x^{a} \times x^{b} \times x^{c} = x^{a+b+c} \][/tex]
Therefore:
[tex]\[
x^{\frac{k-m}{2 m}} \times x^{\frac{m-n}{m n}} \times x^{\frac{n-1}{n l}} = x^{\left( \frac{k-m}{2 m} + \frac{m-n}{m n} + \frac{n-1}{n l} \right)}
\][/tex]
2. Set the Combined Exponent Equal to Zero:
Since [tex]\( x^0 = 1 \)[/tex], for the left-hand side to equal to 1, the exponent sum must be zero:
[tex]\[
\frac{k-m}{2 m} + \frac{m-n}{m n} + \frac{n-1}{n l} = 0
\][/tex]
3. Solve for [tex]\(k\)[/tex]:
Our goal is to find the value of [tex]\(k\)[/tex] that satisfies this relationship. The equation simplifies the following way:
[tex]\[
\frac{k-m}{2 m} = - \left( \frac{m-n}{m n} + \frac{n-1}{n l} \right)
\][/tex]
4. Combine Like Terms:
Combine the fractions on the right-hand side to a common denominator, but preserve each component’s terms clearly:
Let's break it down:
[tex]\[
\frac{m-n}{m n} = \frac{m}{m n} - \frac{n}{m n} = \frac{1}{n} - \frac{1}{m} \quad \text{and} \quad \frac{n-1}{n l} = \frac{n}{n l} - \frac{1}{n l} = \frac{1}{l} - \frac{1}{nl}
\][/tex]
Combine these:
[tex]\[
\frac{1}{n} - \frac{1}{m} + \frac{1}{l} - \frac{1}{nl}
\][/tex]
5. Substitute Back into the Equation:
This would mean:
[tex]\[
\frac{k-m}{2 m} = - \left( \frac{1}{n} - \frac{1}{m} + \frac{1}{l} - \frac{1}{nl} \right)
\][/tex]
6. Isolate [tex]\(k\)[/tex]:
To isolate [tex]\(k\)[/tex], multiply both sides by [tex]\(2m\)[/tex]:
[tex]\[
k - m = -2 m \left( \frac{1}{n} - \frac{1}{m} + \frac{1}{l} - \frac{1}{nl} \right)
\][/tex]
Distribute the [tex]\(-2m\)[/tex]:
[tex]\[
k - m = -2m \left( \frac{1}{n} \right) + 2m \left( \frac{1}{m} \right) - 2m \left( \frac{1}{l} \right) + 2m \left( \frac{1}{nl} \right)
\][/tex]
Combine this to get:
[tex]\[
k - m = -\frac{2m}{n} + 2 - \frac{2m}{l} + \frac{2m}{nl}
\][/tex]
Finally, solve for [tex]\(k\)[/tex]:
[tex]\[
k = m - \frac{2m}{n} + 2 - \frac{2m}{l} + \frac{2m}{nl}
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
k = m - \frac{2m}{n} + 2 - \frac{2m}{l} + \frac{2m}{nl}
\][/tex]
