Algebra

(sqrt(11) - sqrt(2)) / (sqrt(2) + sqrt(11))

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Identify the expression to simplify

We need to simplify this fraction. The denominator has two square root terms added together, so we'll use a technique called rationalizing the denominator.
2

Multiply by the conjugate of denominator

The conjugate of is . Multiplying by this over itself equals 1, so we don't change the value. This eliminates the square roots in the denominator.
3

Expand the numerator using FOIL

We square the numerator: , then , then . Add them together to get .
4

Expand the denominator using difference of squares

The difference of squares pattern says . Here and , so we get . No square roots remain in the denominator!
5

Write the simplified final fraction

Put the expanded numerator over the simplified denominator. There are no common factors between 13, 2, and 9, so this fraction is fully simplified.

Understanding this problem

Learning Insight

Rationalizing the denominator works because of the difference of squares identity: . When and are square roots, squaring them produces whole numbers, clearing the radicals from the denominator. This is a standard algebraic move to rewrite expressions in a cleaner form.

Quick Tip

Whenever you see , immediately multiply top and bottom by . The denominator will always become a plain integer with no radicals.

Common Mistake

Students often forget to also multiply the numerator by the conjugate, or they multiply the numerator by the original denominator instead of the conjugate — both change the value of the expression.