Answer :
To determine which of the given expressions simplifies to [tex]\(1\)[/tex], we'll analyze each one step-by-step.
Given:
1. [tex]\(\left(\frac{m+9}{m-9}\right)\left(\frac{m+9}{m-9}\right)\)[/tex]
2. [tex]\(\left(\frac{m+9}{m-9}\right)\left(\frac{m-9}{m+9}\right)\)[/tex]
3. [tex]\(\left(\frac{m+9}{m-9}\right)\left(\frac{9+m}{9-m}\right)\)[/tex]
4. [tex]\(\left(\frac{m+9}{m-9}\right)\left(\frac{9-m}{9+m}\right)\)[/tex]
### Expression 1:
[tex]\[ \left(\frac{m+9}{m-9}\right) \left(\frac{m+9}{m-9}\right) \][/tex]
Multiply the fractions directly:
[tex]\[ \frac{(m+9)(m+9)}{(m-9)(m-9)} = \frac{(m+9)^2}{(m-9)^2} \][/tex]
This expression simplifies to [tex]\(\frac{(m+9)^2}{(m-9)^2}\)[/tex], which is not equal to [tex]\(1\)[/tex].
### Expression 2:
[tex]\[ \left(\frac{m+9}{m-9}\right) \left(\frac{m-9}{m+9}\right) \][/tex]
Multiply the fractions directly:
[tex]\[ \frac{(m+9)(m-9)}{(m-9)(m+9)} = \frac{(m+9)(m-9)}{(m+9)(m-9)} = 1 \][/tex]
This expression simplifies to [tex]\(1\)[/tex].
### Expression 3:
[tex]\[ \left(\frac{m+9}{m-9}\right) \left(\frac{9+m}{9-m}\right) \][/tex]
Simplify the second term:
[tex]\[ \frac{9+m}{9-m} = \frac{m+9}{-(m-9)} = -\frac{m+9}{m-9} \][/tex]
Now multiply the fractions:
[tex]\[ \left(\frac{m+9}{m-9}\right) \left(-\frac{m+9}{m-9}\right) = -\frac{(m+9)(m+9)}{(m-9)(m-9)} = -\frac{(m+9)^2}{(m-9)^2} \][/tex]
This expression simplifies to [tex]\(-\frac{(m+9)^2}{(m-9)^2}\)[/tex], which is not equal to [tex]\(1\)[/tex].
### Expression 4:
[tex]\[ \left(\frac{m+9}{m-9}\right) \left(\frac{9-m}{9+m}\right) \][/tex]
Simplify the second term:
[tex]\[ \frac{9-m}{9+m} = -\frac{m-9}{m+9} \][/tex]
Now multiply the fractions:
[tex]\[ \left(\frac{m+9}{m-9}\right) \left(-\frac{m-9}{m+9}\right) = -\frac{(m+9)(m-9)}{(m-9)(m+9)} = -1 \][/tex]
This expression simplifies to [tex]\(-1\)[/tex], which is not equal to [tex]\(1\)[/tex].
Thus, after evaluating all the expressions, the one that simplifies to [tex]\(1\)[/tex] is:
[tex]\[ \left( \frac{m+9}{m-9} \right) \left( \frac{m-9}{m+9} \right) \][/tex]
So, the correct answer is the second expression:
[tex]\[ \left(\frac{m+9}{m-9}\right)\left(\frac{m-9}{m+9}\right) \][/tex]
Given:
1. [tex]\(\left(\frac{m+9}{m-9}\right)\left(\frac{m+9}{m-9}\right)\)[/tex]
2. [tex]\(\left(\frac{m+9}{m-9}\right)\left(\frac{m-9}{m+9}\right)\)[/tex]
3. [tex]\(\left(\frac{m+9}{m-9}\right)\left(\frac{9+m}{9-m}\right)\)[/tex]
4. [tex]\(\left(\frac{m+9}{m-9}\right)\left(\frac{9-m}{9+m}\right)\)[/tex]
### Expression 1:
[tex]\[ \left(\frac{m+9}{m-9}\right) \left(\frac{m+9}{m-9}\right) \][/tex]
Multiply the fractions directly:
[tex]\[ \frac{(m+9)(m+9)}{(m-9)(m-9)} = \frac{(m+9)^2}{(m-9)^2} \][/tex]
This expression simplifies to [tex]\(\frac{(m+9)^2}{(m-9)^2}\)[/tex], which is not equal to [tex]\(1\)[/tex].
### Expression 2:
[tex]\[ \left(\frac{m+9}{m-9}\right) \left(\frac{m-9}{m+9}\right) \][/tex]
Multiply the fractions directly:
[tex]\[ \frac{(m+9)(m-9)}{(m-9)(m+9)} = \frac{(m+9)(m-9)}{(m+9)(m-9)} = 1 \][/tex]
This expression simplifies to [tex]\(1\)[/tex].
### Expression 3:
[tex]\[ \left(\frac{m+9}{m-9}\right) \left(\frac{9+m}{9-m}\right) \][/tex]
Simplify the second term:
[tex]\[ \frac{9+m}{9-m} = \frac{m+9}{-(m-9)} = -\frac{m+9}{m-9} \][/tex]
Now multiply the fractions:
[tex]\[ \left(\frac{m+9}{m-9}\right) \left(-\frac{m+9}{m-9}\right) = -\frac{(m+9)(m+9)}{(m-9)(m-9)} = -\frac{(m+9)^2}{(m-9)^2} \][/tex]
This expression simplifies to [tex]\(-\frac{(m+9)^2}{(m-9)^2}\)[/tex], which is not equal to [tex]\(1\)[/tex].
### Expression 4:
[tex]\[ \left(\frac{m+9}{m-9}\right) \left(\frac{9-m}{9+m}\right) \][/tex]
Simplify the second term:
[tex]\[ \frac{9-m}{9+m} = -\frac{m-9}{m+9} \][/tex]
Now multiply the fractions:
[tex]\[ \left(\frac{m+9}{m-9}\right) \left(-\frac{m-9}{m+9}\right) = -\frac{(m+9)(m-9)}{(m-9)(m+9)} = -1 \][/tex]
This expression simplifies to [tex]\(-1\)[/tex], which is not equal to [tex]\(1\)[/tex].
Thus, after evaluating all the expressions, the one that simplifies to [tex]\(1\)[/tex] is:
[tex]\[ \left( \frac{m+9}{m-9} \right) \left( \frac{m-9}{m+9} \right) \][/tex]
So, the correct answer is the second expression:
[tex]\[ \left(\frac{m+9}{m-9}\right)\left(\frac{m-9}{m+9}\right) \][/tex]
