Surds – simplifying, adding, multiplying and rationalising the denominator

What it shows

A surd is a root that cannot be written as a fraction, such as √2. If n = a²b with a² the largest square factor, then √n = a√b: a square of area n is made of a × a squares of area b, so its side is a√b. Only like surds (the same √b) can be added, just like like terms, while √m × √r = √(mr). To rationalise a denominator, multiply top and bottom by √b, or by the conjugate a ∓ √b, since (a + √b)(a − √b) = a² − b is rational.

How to use

On Simplify √n, move the n slider or press Random, then click a square factor to see the grid change; only the largest one finishes in one step. On Add & multiply, choose an Operation and set p, m, q and r, or a, b, c, d and e for expanding. On Rationalise, pick a Form and set k, a and b, then use Next step and compare the decimal check.

Parameters you can change

• Screen Simplify √n, Add & multiply, Rationalise
• Number under the root n 2–200
• Operation Add/subtract: p√m + q√r, Multiply: p√m × q√r, Expand (a + b√c)(d + e√c)
• Coefficient p -6–6
• Number under the root m 1–100
• Coefficient q -6–6
• Number under the root r 1–100
• Expand: a -9–9
• Expand: b -5–5
• Expand: c 2, 3, 5, 6, 7, 10, 11, 13, 14, 15
• Expand: d -9–9
• Expand: e -5–5
• Denominator form k/√b, k/(a√b), k/(a + √b), k/(a − √b)
• Numerator k 1–20
• Number a in the denominator 1–9
• Number b under the root 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23

Questions to explore

1. Why does choosing the square factor 4 for √72 leave the job unfinished, while 36 finishes it in one step?
2. Can √8 + √18 be written as a single surd, and why can √2 + √3 not?
3. Why does multiplying 1/(3 + √5) by (3 − √5)/(3 − √5) leave a denominator with no surd?