Math Problem Statement

3√2 / 4+2√5

Solution

The expression you provided is:

324+25\frac{3\sqrt{2}}{4 + 2\sqrt{5}}

To simplify this, we rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of 4+254 + 2\sqrt{5} is 4254 - 2\sqrt{5}.

  1. Multiply both the numerator and the denominator by 4254 - 2\sqrt{5}:

324+25×425425=32(425)(4+25)(425)\frac{3\sqrt{2}}{4 + 2\sqrt{5}} \times \frac{4 - 2\sqrt{5}}{4 - 2\sqrt{5}} = \frac{3\sqrt{2}(4 - 2\sqrt{5})}{(4 + 2\sqrt{5})(4 - 2\sqrt{5})}

  1. First, simplify the denominator using the difference of squares formula (a+b)(ab)=a2b2(a + b)(a - b) = a^2 - b^2:

(4+25)(425)=42(25)2=1620=4(4 + 2\sqrt{5})(4 - 2\sqrt{5}) = 4^2 - (2\sqrt{5})^2 = 16 - 20 = -4

  1. Now, simplify the numerator:

32(425)=1226103\sqrt{2}(4 - 2\sqrt{5}) = 12\sqrt{2} - 6\sqrt{10}

  1. Putting it all together, we now have:

1226104=32+3102\frac{12\sqrt{2} - 6\sqrt{10}}{-4} = -3\sqrt{2} + \frac{3\sqrt{10}}{2}

So, the simplified expression is:

32+3102-3\sqrt{2} + \frac{3\sqrt{10}}{2}

Would you like further details or have any questions?

Here are five related questions you might find interesting:

  1. How do you rationalize other complex denominators?
  2. What is the difference between a rational and an irrational number?
  3. Can you apply the difference of squares to any radical expression?
  4. What is the general form of a conjugate and how is it useful in simplifications?
  5. How would this simplification change if there were additional radicals in the numerator?

Tip: Always multiply by the conjugate to eliminate square roots from the denominator when rationalizing!

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Math Problem Analysis

Mathematical Concepts

Algebra
Radical Expressions
Rationalization
Difference of Squares

Formulas

Difference of squares: (a + b)(a - b) = a^2 - b^2
Rationalizing denominators using conjugates

Theorems

Difference of Squares Theorem

Suitable Grade Level

Grades 10-12