Math Problem Statement

Rasionalkan bentuk akar √8/4-√6

Solution

We are tasked with rationalizing the expression:

846\frac{\sqrt{8}}{4 - \sqrt{6}}

Step 1: Simplify 8\sqrt{8}

First, simplify 8\sqrt{8}:

8=4×2=22\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}

Thus, the expression becomes:

2246\frac{2\sqrt{2}}{4 - \sqrt{6}}

Step 2: Multiply by the conjugate

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of 464 - \sqrt{6}, which is 4+64 + \sqrt{6}. The conjugate helps eliminate the square root in the denominator.

So, multiply by:

4+64+6\frac{4 + \sqrt{6}}{4 + \sqrt{6}}

This gives:

22(4+6)(46)(4+6)\frac{2\sqrt{2}(4 + \sqrt{6})}{(4 - \sqrt{6})(4 + \sqrt{6})}

Step 3: Apply the difference of squares formula

In the denominator, use the difference of squares formula: (ab)(a+b)=a2b2(a - b)(a + b) = a^2 - b^2, where a=4a = 4 and b=6b = \sqrt{6}:

(46)(4+6)=42(6)2=166=10(4 - \sqrt{6})(4 + \sqrt{6}) = 4^2 - (\sqrt{6})^2 = 16 - 6 = 10

Thus, the expression becomes:

22(4+6)10\frac{2\sqrt{2}(4 + \sqrt{6})}{10}

Step 4: Simplify the numerator

Now distribute 222\sqrt{2} across the terms in the numerator:

22(4+6)=22×4+22×6=82+2122\sqrt{2}(4 + \sqrt{6}) = 2\sqrt{2} \times 4 + 2\sqrt{2} \times \sqrt{6} = 8\sqrt{2} + 2\sqrt{12}

Simplify 12\sqrt{12}:

12=4×3=23\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}

Thus, the numerator becomes:

82+2(23)=82+438\sqrt{2} + 2(2\sqrt{3}) = 8\sqrt{2} + 4\sqrt{3}

Step 5: Final expression

Now, the rationalized expression is:

82+4310\frac{8\sqrt{2} + 4\sqrt{3}}{10}

Simplify by dividing both terms in the numerator by 10:

8210+4310=425+235\frac{8\sqrt{2}}{10} + \frac{4\sqrt{3}}{10} = \frac{4\sqrt{2}}{5} + \frac{2\sqrt{3}}{5}

Thus, the rationalized form of the expression is:

425+235\frac{4\sqrt{2}}{5} + \frac{2\sqrt{3}}{5}

Would you like any more details or have any questions?

Related Questions:

  1. How do you rationalize expressions with cube roots?
  2. What is the general method for simplifying radicals?
  3. Can you explain the difference of squares formula in more detail?
  4. How do you simplify complex fractions involving radicals?
  5. What is the importance of rationalizing denominators?

Tip:

Always look for conjugates when rationalizing denominators with square roots to simplify the process.

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Math Problem Analysis

Mathematical Concepts

Algebra
Radicals
Rationalization
Conjugates
Difference of Squares

Formulas

√a × √b = √(a × b)
(a - b)(a + b) = a^2 - b^2

Theorems

Difference of Squares Theorem
Radical Simplification

Suitable Grade Level

Grades 10-12