If `a^(2) + (1)/(a^(2))= 47 and a ne 0`, find:
`a+ (1)/(a)`

Answer

To solve the problem, we start with the given equation:

**Step 1:** Write down the given equation.
\[ a^2 + \frac{1}{a^2} = 47 \]

**Hint:** Identify the expression you need to manipulate to find \( a + \frac{1}{a} \).

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**Step 2:** Recall the identity for the square of a sum.
We know that:
\[ \left( a + \frac{1}{a} \right)^2 = a^2 + 2 + \frac{1}{a^2} \]

**Hint:** This identity will help us relate \( a^2 + \frac{1}{a^2} \) to \( a + \frac{1}{a} \).

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**Step 3:** Rearrange the identity to express \( a^2 + \frac{1}{a^2} \).
From the identity, we can rearrange it to:
\[ a^2 + \frac{1}{a^2} = \left( a + \frac{1}{a} \right)^2 - 2 \]

**Hint:** This step shows how to express the original equation in terms of the desired quantity.

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**Step 4:** Substitute the value from the original equation into the rearranged identity.
Substituting \( 47 \) for \( a^2 + \frac{1}{a^2} \):
\[ 47 = \left( a + \frac{1}{a} \right)^2 - 2 \]

**Hint:** This substitution will allow you to solve for \( a + \frac{1}{a} \).

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**Step 5:** Solve for \( \left( a + \frac{1}{a} \right)^2 \).
Adding \( 2 \) to both sides gives:
\[ \left( a + \frac{1}{a} \right)^2 = 47 + 2 \]
\[ \left( a + \frac{1}{a} \right)^2 = 49 \]

**Hint:** You are now one step away from finding \( a + \frac{1}{a} \).

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**Step 6:** Take the square root of both sides.
Taking the square root yields:
\[ a + \frac{1}{a} = \pm \sqrt{49} \]
\[ a + \frac{1}{a} = \pm 7 \]

**Hint:** Remember that the square root can yield both positive and negative values.

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**Final Answer:**
Thus, the value of \( a + \frac{1}{a} \) is:
\[ a + \frac{1}{a} = 7 \text{ or } -7 \]

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