Section C 26. If $$ \frac{1+\sqrt{2}}{1-\sqrt{2}}+\frac{1-\sqrt{2}}{1-\sqrt{2}}=a+b \sqrt{2} $$, Find the value of a and b

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UpStudy AI · Accepted Answer

To solve the expression:

$$
\frac{1 + \sqrt{2}}{1 - \sqrt{2}} + \frac{1 - \sqrt{2}}{1 - \sqrt{2}} = a + b\sqrt{2},
$$

let's break it down step by step.

### Step 1: Simplify the Expression
Notice that both fractions have the same denominator:
$$
\frac{1 + \sqrt{2}}{1 - \sqrt{2}} + \frac{1 - \sqrt{2}}{1 - \sqrt{2}} = \frac{(1 + \sqrt{2}) + (1 - \sqrt{2})}{1 - \sqrt{2}}.
$$

Combine the numerators:
$$
\frac{1 + \sqrt{2} + 1 - \sqrt{2}}{1 - \sqrt{2}} = \frac{2}{1 - \sqrt{2}}.
$$

### Step 2: Rationalize the Denominator
To eliminate the square root in the denominator, multiply the numerator and denominator by the conjugate of the denominator $$1 + \sqrt{2}$$:
$$
\frac{2}{1 - \sqrt{2}} \times \frac{1 + \sqrt{2}}{1 + \sqrt{2}} = \frac{2(1 + \sqrt{2})}{(1)^2 - (\sqrt{2})^2} = \frac{2(1 + \sqrt{2})}{1 - 2} = \frac{2(1 + \sqrt{2})}{-1}.
$$

Simplify the expression:
$$
\frac{2(1 + \sqrt{2})}{-1} = -2 - 2\sqrt{2}.
$$

### Step 3: Identify $$a$$ and $$b$$
Comparing the simplified expression to $$a + b\sqrt{2}$$:
$$
-2 - 2\sqrt{2} = a + b\sqrt{2},
$$
we can equate the coefficients:
$$
a = -2 \quad \text{and} \quad b = -2.
$$

**Final Answer:**
$$a = -2$$ and $$b = -2$$.
$$a = -2$$ and $$b = -2$$.

Section C
26. If , Find the value of a and b

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Section C 26. If , Find the value of a and b