Math Problem Statement

Show that (5 - √18) / (1 - √2) can be written in the form a + b√2 where a and b are integers. Show each stage of your working clearly.

Solution

To solve the given problem, we aim to rewrite:

51812\frac{5 - \sqrt{18}}{1 - \sqrt{2}}

in the form a+b2a + b\sqrt{2}, where aa and bb are integers. Let's break it down step by step.


Step 1: Simplify the numerator

The numerator is 5185 - \sqrt{18}. Notice that:

18=92=32.\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}.

So, the numerator becomes:

518=532.5 - \sqrt{18} = 5 - 3\sqrt{2}.


Step 2: Rationalize the denominator

The denominator is 121 - \sqrt{2}. To rationalize it, multiply both the numerator and denominator by the conjugate of the denominator, 1+21 + \sqrt{2}:

532121+21+2.\frac{5 - 3\sqrt{2}}{1 - \sqrt{2}} \cdot \frac{1 + \sqrt{2}}{1 + \sqrt{2}}.


Step 3: Simplify the denominator

Using the difference of squares formula:

(12)(1+2)=12(2)2=12=1.(1 - \sqrt{2})(1 + \sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1.

So, the denominator simplifies to 1-1.


Step 4: Expand the numerator

Now expand (532)(1+2)(5 - 3\sqrt{2})(1 + \sqrt{2}) using the distributive property:

(532)(1+2)=5(1)+5(2)32(1)32(2).(5 - 3\sqrt{2})(1 + \sqrt{2}) = 5(1) + 5(\sqrt{2}) - 3\sqrt{2}(1) - 3\sqrt{2}(\sqrt{2}).

Simplify each term:

=5+52323(2)2.= 5 + 5\sqrt{2} - 3\sqrt{2} - 3(\sqrt{2})^2.

Since (2)2=2(\sqrt{2})^2 = 2, this becomes:

5+52323(2)=5+226.5 + 5\sqrt{2} - 3\sqrt{2} - 3(2) = 5 + 2\sqrt{2} - 6.

Combine like terms:

=1+22.= -1 + 2\sqrt{2}.


Step 5: Divide by the denominator

Now divide the expanded numerator by the denominator, 1-1:

1+221.\frac{-1 + 2\sqrt{2}}{-1}.

Distribute the division:

=11+221=122.= \frac{-1}{-1} + \frac{2\sqrt{2}}{-1} = 1 - 2\sqrt{2}.


Final Answer:

The expression simplifies to:

122,1 - 2\sqrt{2},

where a=1a = 1 and b=2b = -2.


Would you like me to explain any step further or assist with a related problem? Here are some related questions to consider:

  1. What is the general process for rationalizing a denominator with a square root?
  2. How do you simplify square roots such as 18\sqrt{18}?
  3. Why does multiplying by the conjugate work to rationalize denominators?
  4. Can all expressions of this form be rewritten as a+b2a + b\sqrt{2}?
  5. How would the result change if the denominator was 1+21 + \sqrt{2}?

Tip: Always simplify square roots at the earliest step to make calculations easier!

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Math Problem Analysis

Mathematical Concepts

Simplification of algebraic expressions
Rationalization of denominators
Simplifying square roots

Formulas

Difference of squares: (a - b)(a + b) = a^2 - b^2
Square root simplification: √(mn) = √m√n

Theorems

Rationalization of denominators using conjugates

Suitable Grade Level

Grades 9-12