Certainly! Let's go through the problem step by step to find the value of [tex]\( y \)[/tex].
### Given
[tex]\[ x^2 + \frac{1}{x^2} = 7 \][/tex]
and [tex]\( x > 0 \)[/tex].
### Step-by-Step Solution:
1. Calculate [tex]\( y^2 \)[/tex]:
We are given the expression for [tex]\( y^2 \)[/tex].
[tex]\[ y^2 = \left( x + \frac{1}{x} \right)^2 \][/tex]
2. Expand [tex]\( y^2 \)[/tex]:
By expanding the right-hand side, we get:
[tex]\[ \left( x + \frac{1}{x} \right)^2 = x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} \][/tex]
Simplifying this:
[tex]\[ \left( x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2} \][/tex]
3. Substitute the known value:
We know from the problem that:
[tex]\[ x^2 + \frac{1}{x^2} = 7 \][/tex]
Substituting this into our expanded formula for [tex]\( y^2 \)[/tex]:
[tex]\[ y^2 = 7 + 2 \][/tex]
So:
[tex]\[ y^2 = 9 \][/tex]
4. Calculate [tex]\( y \)[/tex]:
To find [tex]\( y \)[/tex], we take the square root of [tex]\( y^2 \)[/tex]:
[tex]\[ y = \sqrt{9} \][/tex]
Therefore:
[tex]\[ y = 3 \][/tex]
### Conclusion
The value of [tex]\( y \)[/tex] is [tex]\( 3 \)[/tex].
### Verification
The result we obtained agrees with the initial calculation we considered:
[tex]\[ y^2 = 9 \][/tex]
[tex]\[ y = 3 \][/tex]
Thus, we have the final answer:
[tex]\[ y = 3 \][/tex]
