Number System & Simplification • Topic 6 of 6

Surds & Indices

An index (exponent) tells how many times a base multiplies itself; a surd is an irrational root such as root 2. The laws of indices (product, quotient, power-of-power, zero and negative exponents) turn ugly expressions into simple ones. To rationalise a denominator like 1/(root a + root b), multiply top and bottom by the conjugate (root a - root b). Compare surds by raising both to the LCM of their root orders.

✅ Solved examples

1. Simplify 2^5 x 2^3 / 2^4.
2^(5+3-4) = 2^4 = 16.
2. Evaluate 27^(2/3).
27^(1/3) = 3, then 3^2 = 9.
3. Rationalise 1/(root 3 - 1).
Multiply by (root 3 + 1): (root 3 + 1)/(3 - 1) = (root 3 + 1)/2.
4. Which is larger, cube root of 4 or square root of 3?
Raise both to power 6: (4^(1/3))^6 = 16; (3^(1/2))^6 = 27. So root 3 is larger.

✏️ Practice — try these, take hints as needed

1. Simplify 3^4 / 3^2.
Subtract exponents.
3^2.
—
9
2. Evaluate 16^(3/4).
16^(1/4) = 2.
2^3.
—
8
3. Value of 2^0 + 3^0?
Any non-zero base ^0 = 1.
1 + 1.
—
2
4. Rationalise 5/(root 5).
Multiply by root 5 / root 5.
5 root 5 / 5.
—
root 5
5. Simplify 9^(1/2) x 8^(1/3).
9^(1/2) = 3.
8^(1/3) = 2.
3 x 2.
6

📝 Topic test — 8 questions

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