Algebra

(sqrt(5) - sqrt(2)) / (sqrt(2) + sqrt(5))

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Identify the conjugate of the denominator

To remove square roots from the denominator, we multiply by its conjugate. The conjugate just flips the sign between the two terms.
2

Multiply numerator and denominator by conjugate

We multiply top and bottom by the same thing, so the value doesn't change. This is like multiplying by 1.
3

Expand the numerator

Use FOIL: , then , then , then . Add like terms.
4

Expand the denominator using difference of squares

The difference of squares pattern says . Squaring each square root removes the radical sign.
5

Write the simplified fraction

Put the expanded numerator over the expanded denominator.
6

Rewrite by distributing the negative sign

Dividing by is the same as flipping the signs on top and making the denominator positive. This is the standard simplified form.

Understanding this problem

Learning Insight

Multiplying by a conjugate works because of the difference of squares identity: . When and are square roots, squaring them produces whole numbers, so all radicals in the denominator disappear. This technique is called 'rationalizing the denominator.'

Quick Tip

Whenever you see a denominator with two square root terms added or subtracted, immediately think 'conjugate.' Just flip the sign between the terms and multiply top and bottom by it.

Common Mistake

A common mistake is multiplying the denominator by the conjugate of the NUMERATOR instead of the conjugate of the DENOMINATOR, or forgetting to also multiply the numerator — either error changes the value of the expression.