Topics

Exponents, roots and surds

Covers AS91261

The index laws

The standard requires fractional and negative exponents, not just whole numbers. All of it rests on four rules:

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

$\frac{a^m}{a^n} = a^{m-n}$

$\left(a^m\right)^n = a^{mn}$

$(ab)^n = a^nb^n$

Multiply — add the powers. Divide — subtract. Power of a power — multiply.
The bases must match for the first two. $2^3 \times 3^2$ cannot be combined.
$x^5 \times x^3 = x^8$ , $\dfrac{x^7}{x^2} = x^5$ , $\left(x^3\right)^4 = x^{12}$ , $(2x)^4 = 16x^4$ .

Zero and negative exponents

$a^0 = 1 \qquad a^{-n} = \frac{1}{a^n}$

Anything to the power zero is 1 (except $0^0$ ). This follows from $\frac{a^n}{a^n} = a^0$ and also $= 1$ .
A negative power means a reciprocal, not a negative number. $2^{-3} = \frac{1}{8}$ , which is positive.
$\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n$ — a negative power flips the fraction.

Fractional exponents

$a^{1/n} = \sqrt[n]{a} \qquad a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$

The bottom of the fraction is the root; the top is the power.
$x^{1/2} = \sqrt{x}$ , $x^{1/3} = \sqrt[3]{x}$ , $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$ .
Take the root first, then the power — the numbers stay much smaller that way. $\left(\sqrt[3]{8}\right)^2 = 4$ is easier than $\sqrt[3]{64} = 4$ .
The 2025 report lists "converted from surd form to index form" and "simplified indices" as Achieved behaviours, so this conversion is expected to be automatic.

Combining them

$x^{-1/2} = \dfrac{1}{\sqrt{x}}$ , $\dfrac{1}{x^{-3}} = x^3$ , $\left(\dfrac{16}{9}\right)^{-1/2} = \left(\dfrac{9}{16}\right)^{1/2} = \dfrac{3}{4}$ .
The general method for a messy expression:
- Convert every root to a fractional power and every reciprocal to a negative power.
- Apply the index laws.
- Convert back if the question wants surd form.

Surds

A surd is an irrational root left in exact form, such as $\sqrt{2}$ .
Simplify by extracting the largest perfect square:

$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$

Multiply: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ , so $2\sqrt{3} \times 4\sqrt{6} = 8\sqrt{18} = 24\sqrt{2}$ .
Add only like surds: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$ , but $3\sqrt{5} + 2\sqrt{3}$ does not combine.
$\sqrt{a+b}$ is not $\sqrt{a} + \sqrt{b}$ . Roots distribute over multiplication only.

Rationalising a denominator

Multiply top and bottom by the surd, or by the conjugate for a two-term denominator:

$\frac{6}{\sqrt{2}} = \frac{6\sqrt{2}}{2} = 3\sqrt{2}$

$\frac{4}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}} = \frac{4(3-\sqrt{5})}{9-5} = 3 - \sqrt{5}$

The conjugate works because $(a+\sqrt{b})(a-\sqrt{b}) = a^2 - b$ , a difference of two squares that removes the root.

Solving equations with fractional powers

Raise both sides to the reciprocal power to undo it.
$x^{2/3} = 9$ : raise both sides to the power $\frac{3}{2}$ , giving $x = 9^{3/2} = 27$ .
Check for a second solution. An even root introduces a $\pm$ ; an odd root does not. $x^{2} = 9$ has two solutions, $x^3 = 8$ has one.

Tips and Tricks

Convert everything to index form before you start. The report names "failed to manipulate expressions that involved surds, indices, or logs" as a Not Achieved behaviour, and almost every instance traces to working with a root still written as a root.

A negative power is a reciprocal, not a negative number. $3^{-2} = \frac{1}{9}$ , never $-9$ and never $-\frac{1}{9}$ .

In $a^{m/n}$ , the denominator is the root. $x^{3/2}$ is the square root cubed, not the cube root squared. Getting this backwards changes the answer.

Take the root before the power when evaluating. $32^{3/5} = \left(\sqrt[5]{32}\right)^3 = 2^3 = 8$ is trivial; $\sqrt[5]{32768}$ is not.

Bases must match before you add or subtract powers. If they do not, rewrite one — $4^x = \left(2^2\right)^x = 2^{2x}$ is the standard move, and it is exactly what exponential equations need.

Leave answers exact — $5\sqrt{2}$ , not 7.07 — unless a decimal is requested. The specification asks for "simplest algebraic form".

Worked ExampleSimplifying with fractional and negative indices

Simplify $\dfrac{\left(8x^6\right)^{2/3}}{4x^{-2}}$ , giving your answer with positive indices only.

Step 1 — Deal with the numerator's outer power

The power $\frac{2}{3}$ applies to everything inside the bracket — both the 8 and the $x^6$ :

$\left(8x^6\right)^{2/3} = 8^{2/3} \times \left(x^6\right)^{2/3}$

Step 2 — Evaluate the numerical part

Take the root first, then the power:

$8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$

Step 3 — Evaluate the algebraic part

Power of a power — multiply the indices:

$\left(x^6\right)^{2/3} = x^{6 \times 2/3} = x^{4}$

So the numerator is $4x^4$ .

Step 4 — Write the whole fraction

$\frac{4x^4}{4x^{-2}}$

Step 5 — Simplify the numbers

$\frac{4}{4} = 1$

Step 6 — Simplify the powers of $x$

Dividing subtracts the indices. Take care with the double negative:

$\frac{x^4}{x^{-2}} = x^{4 - (-2)} = x^{4+2} = x^{6}$

Step 7 — Combine

$\frac{\left(8x^6\right)^{2/3}}{4x^{-2}} = x^6$

The answer has a positive index, as required.

Step 8 — Check with a numerical value

Substitute $x = 2$ into the original expression:

Numerator: $\left(8 \times 64\right)^{2/3} = 512^{2/3} = \left(\sqrt[3]{512}\right)^2 = 8^2 = 64$
Denominator: $4 \times 2^{-2} = 4 \times \frac{1}{4} = 1$
Original value: $\frac{64}{1} = 64$

And the simplified form: $x^6 = 2^6 = 64$ ✓

Exponents, roots and surds

Covers AS91261

The index laws

The standard requires fractional and negative exponents, not just whole numbers. All of it rests on four rules:

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

$\frac{a^m}{a^n} = a^{m-n}$

$\left(a^m\right)^n = a^{mn}$

$(ab)^n = a^nb^n$

Multiply — add the powers. Divide — subtract. Power of a power — multiply.
The bases must match for the first two. $2^3 \times 3^2$ cannot be combined.
$x^5 \times x^3 = x^8$ , $\dfrac{x^7}{x^2} = x^5$ , $\left(x^3\right)^4 = x^{12}$ , $(2x)^4 = 16x^4$ .

Zero and negative exponents

$a^0 = 1 \qquad a^{-n} = \frac{1}{a^n}$

Anything to the power zero is 1 (except $0^0$ ). This follows from $\frac{a^n}{a^n} = a^0$ and also $= 1$ .
A negative power means a reciprocal, not a negative number. $2^{-3} = \frac{1}{8}$ , which is positive.
$\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n$ — a negative power flips the fraction.

Fractional exponents

$a^{1/n} = \sqrt[n]{a} \qquad a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$

The bottom of the fraction is the root; the top is the power.
$x^{1/2} = \sqrt{x}$ , $x^{1/3} = \sqrt[3]{x}$ , $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$ .
Take the root first, then the power — the numbers stay much smaller that way. $\left(\sqrt[3]{8}\right)^2 = 4$ is easier than $\sqrt[3]{64} = 4$ .
The 2025 report lists "converted from surd form to index form" and "simplified indices" as Achieved behaviours, so this conversion is expected to be automatic.

Combining them

$x^{-1/2} = \dfrac{1}{\sqrt{x}}$ , $\dfrac{1}{x^{-3}} = x^3$ , $\left(\dfrac{16}{9}\right)^{-1/2} = \left(\dfrac{9}{16}\right)^{1/2} = \dfrac{3}{4}$ .
The general method for a messy expression:
- Convert every root to a fractional power and every reciprocal to a negative power.
- Apply the index laws.
- Convert back if the question wants surd form.

Surds

A surd is an irrational root left in exact form, such as $\sqrt{2}$ .
Simplify by extracting the largest perfect square:

$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$

Multiply: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ , so $2\sqrt{3} \times 4\sqrt{6} = 8\sqrt{18} = 24\sqrt{2}$ .
Add only like surds: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$ , but $3\sqrt{5} + 2\sqrt{3}$ does not combine.
$\sqrt{a+b}$ is not $\sqrt{a} + \sqrt{b}$ . Roots distribute over multiplication only.

Rationalising a denominator

Multiply top and bottom by the surd, or by the conjugate for a two-term denominator:

$\frac{6}{\sqrt{2}} = \frac{6\sqrt{2}}{2} = 3\sqrt{2}$

$\frac{4}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}} = \frac{4(3-\sqrt{5})}{9-5} = 3 - \sqrt{5}$

The conjugate works because $(a+\sqrt{b})(a-\sqrt{b}) = a^2 - b$ , a difference of two squares that removes the root.

Solving equations with fractional powers

Raise both sides to the reciprocal power to undo it.
$x^{2/3} = 9$ : raise both sides to the power $\frac{3}{2}$ , giving $x = 9^{3/2} = 27$ .
Check for a second solution. An even root introduces a $\pm$ ; an odd root does not. $x^{2} = 9$ has two solutions, $x^3 = 8$ has one.

Tips and Tricks

A negative power is a reciprocal, not a negative number. $3^{-2} = \frac{1}{9}$ , never $-9$ and never $-\frac{1}{9}$ .

In $a^{m/n}$ , the denominator is the root. $x^{3/2}$ is the square root cubed, not the cube root squared. Getting this backwards changes the answer.

Take the root before the power when evaluating. $32^{3/5} = \left(\sqrt[5]{32}\right)^3 = 2^3 = 8$ is trivial; $\sqrt[5]{32768}$ is not.

Bases must match before you add or subtract powers. If they do not, rewrite one — $4^x = \left(2^2\right)^x = 2^{2x}$ is the standard move, and it is exactly what exponential equations need.

Leave answers exact — $5\sqrt{2}$ , not 7.07 — unless a decimal is requested. The specification asks for "simplest algebraic form".

Worked ExampleSimplifying with fractional and negative indices

Simplify $\dfrac{\left(8x^6\right)^{2/3}}{4x^{-2}}$ , giving your answer with positive indices only.

Step 1 — Deal with the numerator's outer power

The power $\frac{2}{3}$ applies to everything inside the bracket — both the 8 and the $x^6$ :

$\left(8x^6\right)^{2/3} = 8^{2/3} \times \left(x^6\right)^{2/3}$

Step 2 — Evaluate the numerical part

Take the root first, then the power:

$8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$

Step 3 — Evaluate the algebraic part

Power of a power — multiply the indices:

$\left(x^6\right)^{2/3} = x^{6 \times 2/3} = x^{4}$

So the numerator is $4x^4$ .

Step 4 — Write the whole fraction

$\frac{4x^4}{4x^{-2}}$

Step 5 — Simplify the numbers

$\frac{4}{4} = 1$

Step 6 — Simplify the powers of $x$

Dividing subtracts the indices. Take care with the double negative:

$\frac{x^4}{x^{-2}} = x^{4 - (-2)} = x^{4+2} = x^{6}$

Step 7 — Combine

$\frac{\left(8x^6\right)^{2/3}}{4x^{-2}} = x^6$

The answer has a positive index, as required.

Step 8 — Check with a numerical value

Substitute $x = 2$ into the original expression:

Numerator: $\left(8 \times 64\right)^{2/3} = 512^{2/3} = \left(\sqrt[3]{512}\right)^2 = 8^2 = 64$
Denominator: $4 \times 2^{-2} = 4 \times \frac{1}{4} = 1$
Original value: $\frac{64}{1} = 64$

And the simplified form: $x^6 = 2^6 = 64$ ✓

Exponents, roots and surds

The index laws

Zero and negative exponents

Fractional exponents

Combining them

Surds

Rationalising a denominator

Solving equations with fractional powers

Step 1 — Deal with the numerator's outer power

Step 2 — Evaluate the numerical part

Step 3 — Evaluate the algebraic part

Step 4 — Write the whole fraction

Step 5 — Simplify the numbers

Step 6 — Simplify the powers of xxx

Step 7 — Combine

Step 8 — Check with a numerical value

Exponents, roots and surds

The index laws

Zero and negative exponents

Fractional exponents

Combining them

Surds

Rationalising a denominator

Solving equations with fractional powers

Step 1 — Deal with the numerator's outer power

Step 2 — Evaluate the numerical part

Step 3 — Evaluate the algebraic part

Step 4 — Write the whole fraction

Step 5 — Simplify the numbers

Step 6 — Simplify the powers of xxx

Step 7 — Combine

Step 8 — Check with a numerical value

Step 6 — Simplify the powers of $x$

Step 6 — Simplify the powers of $x$