Math Problem Statement

The rationalising factor of 4 root 5 - 1

Solution

To rationalize the expression 4514\sqrt{5} - 1, we can multiply the numerator and denominator by the conjugate of the expression. The conjugate of 4514\sqrt{5} - 1 is 45+14\sqrt{5} + 1.

So, we multiply:

145145+145+1=45+1(451)(45+1)\frac{1}{4\sqrt{5} - 1} \cdot \frac{4\sqrt{5} + 1}{4\sqrt{5} + 1} = \frac{4\sqrt{5} + 1}{(4\sqrt{5} - 1)(4\sqrt{5} + 1)}

The denominator simplifies using the difference of squares:

(45)212=801=79(4\sqrt{5})^2 - 1^2 = 80 - 1 = 79

Thus, the rationalized form is:

45+179\frac{4\sqrt{5} + 1}{79}

Would you like further details on any step, or do you have any questions? Here are some related questions you might find interesting:

  1. What is the significance of rationalizing denominators in mathematics?
  2. Can you provide another example of rationalizing a different expression?
  3. How does rationalizing affect the value of an expression?
  4. What is the general form of a conjugate for a binomial expression?
  5. Why do we prefer rationalizing expressions in certain mathematical contexts?

Tip: Always use the conjugate to simplify expressions involving square roots or other irrational numbers.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rationalizing Expressions
Conjugates

Formulas

Rationalizing formula: \( \frac{a}{b} \cdot \frac{c}{c} = \frac{ac}{bc} \)
Difference of squares: \( a^2 - b^2 = (a-b)(a+b) \)

Theorems

Conjugate theorem

Suitable Grade Level

Grades 9-12