https://file.aiquickdraw.com/mathbot/file/1729779093344fig2eyfy.jpg

We are given that \( a = 9 - 4\sqrt{5} \), and we need to find the value of:

\[
a^2 + \frac{1}{a^2}
\]

### Step 1: Compute \( a^2 \)

First, let's calculate \( a^2 \):

\[
a = 9 - 4\sqrt{5}
\]

So,

\[
a^2 = (9 - 4\sqrt{5})^2
\]

Expanding this expression using the identity \( (x - y)^2 = x^2 - 2xy + y^2 \):

\[
a^2 = 9^2 - 2 \times 9 \times 4\sqrt{5} + (4\sqrt{5})^2
\]

Calculating each term:

\[
9^2 = 81
\]
\[
2 \times 9 \times 4\sqrt{5} = 72\sqrt{5}
\]
\[
(4\sqrt{5})^2 = 16 \times 5 = 80
\]

Thus,

\[
a^2 = 81 - 72\sqrt{5} + 80 = 161 - 72\sqrt{5}
\]

### Step 2: Find \( \frac{1}{a} \) and \( \frac{1}{a^2} \)

Now, we need to find \( \frac{1}{a^2} \). To do that, let's first calculate \( \frac{1}{a} \). The reciprocal of \( a = 9 - 4\sqrt{5} \) can be found by multiplying the numerator and denominator by the conjugate \( 9 + 4\sqrt{5} \):

\[
\frac{1}{a} = \frac{1}{9 - 4\sqrt{5}} \times \frac{9 + 4\sqrt{5}}{9 + 4\sqrt{5}}
\]

In the denominator, we use the difference of squares formula:

\[
(9 - 4\sqrt{5})(9 + 4\sqrt{5}) = 9^2 - (4\sqrt{5})^2 = 81 - 80 = 1
\]

Thus,

\[
\frac{1}{a} = 9 + 4\sqrt{5}
\]

Now, squaring both sides:

\[
\frac{1}{a^2} = (9 + 4\sqrt{5})^2
\]

Using the same binomial expansion method:

\[
(9 + 4\sqrt{5})^2 = 9^2 + 2 \times 9 \times 4\sqrt{5} + (4\sqrt{5})^2
\]
\[
= 81 + 72\sqrt{5} + 80 = 161 + 72\sqrt{5}
\]

### Step 3: Add \( a^2 \) and \( \frac{1}{a^2} \)

Now, we add \( a^2 \) and \( \frac{1}{a^2} \):

\[
a^2 + \frac{1}{a^2} = (161 - 72\sqrt{5}) + (161 + 72\sqrt{5}) = 161 + 161 = 322
\]

### Final Answer:
The value of \( a^2 + \frac{1}{a^2} \) is \( \boxed{322} \).

Would you like further details on any step?

Here are five related questions to explore:
1. How do we simplify expressions involving square roots in a binomial?
2. What is the purpose of multiplying by the conjugate in rationalizing the denominator?
3. Could you explain the difference of squares formula and when it's used?
4. How would the result change if \( a \) had been a different form, such as \( 9 + 4\sqrt{5} \)?
5. What other methods exist for simplifying such expressions involving radicals?

**Tip:** When dealing with square roots and fractions, conjugates are often a powerful tool to simplify expressions.

If a = 9 - 4√5, find the value of a^2 + 1/a^2.

This algebra problem demonstrates how to find the value of a^2 + 1/a^2 when a = 9 - 4√5. It involves binomial expansion, rationalizing denominators, and the difference of squares theorem. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation