Radical expressions: roots as fractional powers

A root is a power in disguise

If n is a positive integer greater than 1, any real x with x = ⁿ√a is the nth root of a. The sign √ is the radical, n is its index and a is the radicand.

Your book writes every root as a power: ⁿ√a = a^(1/n). Once a root is a power, the laws of indices do all the work.

Multiply powers of the same base: add the indices. Divide: subtract them. A power of a power: multiply them. Anything (non-zero) to the power 0 is 1.

Turn the root into a power, split the product, then multiply the indices. That is the whole of Example 7.

Key terms

Radical

Marks a root.

Index

Which root is taken.

Radicand

What the root is taken of.

Exponential form

Lets the laws of indices handle roots.

Short questions with model answers

1. Q1. Example 7 (i): simplify ⁴√(16x⁴y⁸).

   • (16x⁴y⁸)^(1/4)
   • (16)^(1/4) (x⁴)^(1/4) (y⁸)^(1/4)
   • 2^(4·1/4) × x^(4·1/4) × y^(8·1/4) = 2xy²

   2xy².

2. Q2. Example 7 (ii): simplify ∛(27x⁶y⁹z³).

   • (3³)^(1/3) (x⁶)^(1/3) (y⁹)^(1/3) (z³)^(1/3)
   • 3^(3·1/3) x^(6·1/3) y^(9·1/3) z^(3·1/3)

   3x²y³z.

Common mistakes

• ✗ Writing (x⁴)^(1/4) = x^(4 + 1/4).

  ✓ A power of a power multiplies: x^(4 × 1/4) = x.

• ✗ Treating (64)^(−4/3) as a negative number.

  ✓ A negative index means a reciprocal: a^(−m) = 1/aᵐ, so the answer is 1/256.

• ✗ Leaving 16 outside the fourth root: ⁴√(16x⁴y⁸) = 16xy².

  ✓ The root acts on every factor: ⁴√16 = 2.

MCQs

1. 1. In ∛5, the index is:

   1. (a) 5
   2. (b) 1/3
   3. (c) 3
   4. (d) 15
   Show answer

   (c) n = 3 is the index; 5 is the radicand (p. 11).

2. 2. ⁴√(16x⁴y⁸) =

   1. (a) 2xy²
   2. (b) 4xy²
   3. (c) 2x⁴y⁸
   4. (d) 16xy²
   Show answer

   (a) Example 7 (i), p. 12.

3. 3. (64)^(−4/3) =

   1. (a) −256
   2. (b) 256
   3. (c) 1/64
   4. (d) 1/256
   Show answer

   (d) 1/(4³)^(4/3) = 1/4⁴ = 1/256 (p. 12).

4. 4. aᵐ / aⁿ equals:

   1. (a) a^(m+n)
   2. (b) a^(m−n)
   3. (c) a^(mn)
   4. (d) a^(m/n)
   Show answer

   (b) Dividing subtracts the indices (p. 11).

5. 5. If 2ˣ × 8ˣ = 64, then x =

   1. (a) 3/2
   2. (b) 2
   3. (c) 3
   4. (d) 1
   Show answer

   (a) 2ˣ·2³ˣ = 2⁴ˣ = 2⁶, so 4x = 6, x = 3/2 (Review Exercise 1 vii).

Quick revision

• ⁿ√a = a^(1/n): the radical, its index n and the radicand a.
• Multiply: add indices; divide: subtract; power of a power: multiply.
• A negative index gives a reciprocal: (64)^(−4/3) = 1/256.