Nested roots: √(a ± 2√b)
Get rid of a root under a root: write the expression as the square of a sum or a difference and find the numbers from their sum and product.
- Russia: grade 8
Guide
This is an advanced topic: in the Russian programme it belongs to the advanced level of Grade 8 ("transforming expressions with square roots"). It can be skipped in an ordinary lesson.
An expression a + 2√b is sometimes a square: (√x + √y)2 = x + y + 2√(xy). So find two numbers with sum a and product b. Then √(a + 2√b) = √x + √y, and for a difference √(a − 2√b) = √x − √y when x > y.
Another way: look for the answer as m + n√3 and square it: (m + n√3)2 = m2 + 3n2 + 2mn√3. Match the rational parts and the parts with the root.
In this studio you find the number with a table: type a value, the second column works out the other number from the product and the last column checks the sum.