https://file.aiquickdraw.com/mathbot/file/1727802569724yw4m0lca.jpg

We are given that \( a = 2 + \sqrt{3} \), and we are asked to find the value of \( a - \frac{1}{a} \).

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

We are given \( a = 2 + \sqrt{3} \). To find \( \frac{1}{a} \), we rationalize the denominator by multiplying both the numerator and the denominator by \( 2 - \sqrt{3} \), the conjugate of \( 2 + \sqrt{3} \).

\[
\frac{1}{a} = \frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})}
\]

Now, simplify the denominator:

\[
(2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1
\]

Thus,

\[
\frac{1}{a} = 2 - \sqrt{3}
\]

### Step 2: Compute \( a - \frac{1}{a} \)

Now we subtract \( \frac{1}{a} \) from \( a \):

\[
a - \frac{1}{a} = (2 + \sqrt{3}) - (2 - \sqrt{3})
\]

Simplify:

\[
a - \frac{1}{a} = 2 + \sqrt{3} - 2 + \sqrt{3} = 2\sqrt{3}
\]

### Final Answer:

\[
a - \frac{1}{a} = 2\sqrt{3}
\]

Would you like any further explanation or details on this? Let me know if you have any questions.

Here are five related questions you might explore:

1. What is the value of \( a + \frac{1}{a} \) when \( a = 2 + \sqrt{3} \)?
2. How do you find the square of \( a \), given \( a = 2 + \sqrt{3} \)?
3. What happens if you change \( a \) to \( 2 - \sqrt{3} \) and calculate \( a - \frac{1}{a} \)?
4. How does rationalizing the denominator help in simplifying expressions?
5. How would the problem change if \( a \) were \( 2 + \sqrt{5} \)?

**Tip**: Rationalizing the denominator is often used to simplify expressions involving square roots. This technique can help transform complex expressions into simpler ones.

If a = 2 + √3, then find the value of a - 1/a.

Learn how to simplify the expression a - 1/a when a = 2 + √3 by rationalizing the denominator and using basic algebraic techniques. The step-by-step process is ideal 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