Keep the exact value.

A power counts repeated multiplication. Multiplying powers with the same base adds their indices; dividing subtracts them. A negative index means a reciprocal, not a negative answer. For a positive base, a fractional index describes a root.

A square root is the non-negative number whose square gives the original value. Pull perfect-square factors out of a radical: $\sqrt{75}$ = $\sqrt{25\times3}$ = 5 $\sqrt{3}$ . Combine only like radicals. This surd work extends the Year 10 foundations towards senior Methods.

INTERACTIVE MODEL

Try it. Watch it change.

Area / square units8

\sqrt{8}

= 2

\sqrt{2}

≈ 2.8284

A square’s side length is the positive square root of its area. Exact surd form avoids rounding.

Explore: Change the square’s area. Compare its exact side length with its decimal approximation.

Product of powers

a^m \times a^n = a^{m+n}

Negative exponent

a^{-n} = \frac{1}{a^n} \qquad (a\ne0)

Simplifying a surd

\sqrt{a^2b}=|a|\sqrt b \qquad (b\ge0)

WORKED EXAMPLE

Simplify $\sqrt{48}$ + $\sqrt{12}$ without rounding.

$\sqrt{48}$ = 4 $\sqrt{3}$ and $\sqrt{12}$ = 2 $\sqrt{3}$ .
Add the coefficients: 6 $\sqrt{3}$ .

Assumed knowledge

Multiplication, factors and positive square numbers.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Use integer and fractional powers.
Simplify square roots and distinguish exact values from approximations.

Year 10 preparation ↗

READY TO TRY IT?

Keep the exact value.

Simplify 48\sqrt{48}48​ + 12\sqrt{12}12​ without rounding.

Put the idea to work.

Simplify $\sqrt{48}$ + $\sqrt{12}$ without rounding.