Year 9 Maths Core Indices and Scientific Notation

Mixed Index Law Questions

Q: What order do you use the index laws in?

Clear the brackets, replace zero indices by 1, multiply, divide, then rewrite any negative index so the answer has positive indices only. In a chain with no fraction line, multiply and divide from left to right.

Q: What does an index of one half mean?

An index of one half means the square root. 25 to the power one half is the square root of 25, which is 5. An index of one third means the cube root.

Q: What happens when you divide by a negative power?

Subtracting a negative index adds. x to the 5 divided by x to the negative 3 is x to the 5 plus 3, which is x to the 8.

20 practice questions 2 video lessons Theory + worked examples

Mixed Index Law Questions puts every index law to work in the same expressions, including zero and negative indices. This NSW Year 9 Core maths subtopic builds on NESA outcome MA5‑IND‑C‑01, and a few questions go further, reading a fractional index as a root.

You will apply two or three laws in one expression, as in \((3b^2)^3 \times b = 27b^7\), clear a bracket that carries a negative index, combine a negative index with a multiplication, and rewrite answers such as \(\frac{10a^2b}{5a^4b^3} = \frac{2}{a^2b^2}\) with positive indices. You will also spot which one-step simplification is correct, simplify long quotients with several pronumerals, and read \(a^{\frac{1}{2}}\) as a square root and \(\sqrt[3]{64k^3}\) as \(4k\). It is part of Year 9 Core Indices and Scientific Notation.

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Theory

Mixed index law questions use several laws in one expression. This Year 9 maths page gives a working order for long questions: brackets, zero indices, multiply, divide, then make every index positive. It also shows how a fractional index means a root.

A mixed question needs more than one index law. The trick is doing them in a sensible order, one short line at a time.

The laws you combine: multiply (add the indices), divide (subtract them), power of a power (multiply them), the zero index (\(a^0 = 1\)) and the negative index (\(a^{-n} = \dfrac{1}{a^n}\)).

"Give your answer with positive indices" means no index in the answer may be negative.

A fractional index is a root: \(a^{1/2} = \sqrt{a}\) and \(a^{1/3} = \sqrt[3]{a}\). A few mixed questions use fractional indices.

Take the laws in this order, one short line each, and a long expression comes apart step by step.

A fractional index is a root: \(a^{1/2} = \sqrt{a}\) is the side of a square of area \(a\), and \(a^{1/3} = \sqrt[3]{a}\) is the edge of a cube of volume \(a\).

The laws, for a base \(a\) that is not \(0\):

\[a^m \times a^n = a^{m+n} \qquad a^m \div a^n = a^{m-n} \qquad (a^m)^n = a^{mn}\]

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

\[a^0 = 1 \qquad a^{-n} = \dfrac{1}{a^n} \qquad \left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n}\]

a^{- n} = \frac{1}{a^{n}}

Roots as indices:

\[a^{1/2} = \sqrt{a} \qquad a^{1/3} = \sqrt[3]{a}\]

a^{1 / 2} = \sqrt{a}

Roots of a product. Take the root of each factor: \(\sqrt[3]{8m^6} = 2m^2\), because \(2^3 = 8\) and \((m^2)^3 = m^6\).

Only one letter asked for? If a question asks for just the index of one letter, follow that letter through each step and ignore the rest.

Dividing by a negative index. \(x^5 \div x^{-3} = x^{5-(-3)} = x^8\). Subtracting a negative adds.

How to simplify a mixed index question

Clear the brackets. Raise every factor inside. A fraction to a negative power flips first.
Replace zero indices by \(1\).
Multiply on the top and on the bottom separately: numbers, then add the indices.
Divide: numbers, then subtract the indices of each letter.
Tidy: move any negative index across the fraction line so every index is positive.

Example 1 — Brackets and negative indices

Simplify, with positive indices
(i) \((2x^3)^4 \times 3x\)
(ii) \(6y^{-7} \times 5y^3\)

Solution

(i) Clear the bracket, raising both factors. Then multiply.

\((2x^3)^4\)	\(=\)	\(2^4 x^{12} = 16x^{12}\)
\(16x^{12} \times 3x^1\)	\(=\)	\(\boldsymbol{48x^{13}}\)

(ii) Multiply the numbers and add the indices. Adding a negative is fine.

\(6y^{-7} \times 5y^3\)	\(=\)	\(30y^{-7+3}\)
\(\)	\(=\)	\(30y^{-4}\)
\(\)	\(=\)	\(\boldsymbol{\dfrac{30}{y^4}}\)

6 y^{- 7} \times 5 y^{3} = \frac{30}{y^{4}}

Only the \(y^4\) moves under the line. The \(30\) stays on top.

Example 2 — A quotient with positive indices

Simplify \(\dfrac{12a^5b^3}{18a^2b^7}\), giving your answer with positive indices.

Solution

Numbers first, then each letter on its own.

\(\dfrac{12}{18}\)	\(=\)	\(\dfrac{2}{3}\)
\(a^5 \div a^2\)	\(=\)	\(a^3\)
\(b^3 \div b^7\)	\(=\)	\(b^{-4}\)

Move the negative index underneath to make it positive.

\(\dfrac{12a^5b^3}{18a^2b^7}\)	\(=\)	\(\dfrac{2a^3}{3} \times \dfrac{1}{b^4}\)
\(\)	\(=\)	\(\boldsymbol{\dfrac{2a^3}{3b^4}}\)

\frac{12 a^{5} b^{3}}{18 a^{2} b^{7}} = \frac{2 a^{3}}{3 b^{4}}

There are more \(b\)s on the bottom, so the \(b\)s finish on the bottom.

Example 3 — Negative and fractional powers

Simplify
(i) \(\left(\dfrac{5m^2}{n}\right)^{-2}\)
(ii) \(\left(\dfrac{9}{a^4}\right)^{-1/2}\)

Solution

(i) Flip the fraction to make the index positive. Then raise the top and the bottom.

\(\left(\dfrac{5m^2}{n}\right)^{-2}\)	\(=\)	\(\left(\dfrac{n}{5m^2}\right)^{2}\)
\(\)	\(=\)	\(\boldsymbol{\dfrac{n^2}{25m^4}}\)

(ii) Flip it the same way. An index of \(\dfrac{1}{2}\) means a square root.

\(\left(\dfrac{9}{a^4}\right)^{-1/2}\)	\(=\)	\(\left(\dfrac{a^4}{9}\right)^{1/2}\)
\(\)	\(=\)	\(\dfrac{a^{4 \times 1/2}}{9^{1/2}}\)
\(\)	\(=\)	\(\boldsymbol{\dfrac{a^2}{3}}\)

{(\frac{9}{a^{4}})}^{- 1 / 2} = \frac{a^{2}}{3}

Check: \(\sqrt{9} = 3\), and \(a^2\) squared is \(a^4\).

Example 4 — Top, bottom, then divide

Simplify \(\dfrac{(3p^2q)^3 \times 2pq^4}{9p^4q^5}\).

Solution

Clear the bracket. All three factors are cubed.

\((3p^2q)^3\)

\(=\)

\(27p^6q^3\)

Multiply out the top: numbers, then add the indices.

\(27p^6q^3 \times 2pq^4\)

\(=\)

\(54p^7q^7\)

Now divide by the bottom.

\(54 \div 9\)	\(=\)	\(6\)
\(p^7 \div p^4\)	\(=\)	\(p^3\)
\(q^7 \div q^5\)	\(=\)	\(q^2\)
\(\text{answer}\)	\(=\)	\(\boldsymbol{6p^3q^2}\)

\frac{{(3 p^{2} q)}^{3} \times 2 p q^{4}}{9 p^{4} q^{5}} = 6 p^{3} q^{2}

Common pitfalls

Clear brackets first. The number inside a bracket is raised too. \((3a^2)^2 = 9a^4\), and every later line depends on it.

Finish with positive indices. \(4x^{-2}\) is not finished. Write it as \(\dfrac{4}{x^2}\).

Subtracting a negative index adds. \(y^2 \div y^{-5} = y^{7}\), not \(y^{-3}\).

A one-step shortcut needs checking. If someone simplified in one step, check which law they used. Multiplying adds indices; a power of a power multiplies them.

Frequently asked questions

What order do you use the index laws in?

Clear the brackets, replace zero indices by \(1\), multiply, divide, then rewrite any negative index so the answer has positive indices only. In a chain with no fraction line, multiply and divide from left to right.

How do you write an answer with positive indices?

Move any part with a negative index to the other side of the fraction line and make the index positive: \(30y^{-4} = \dfrac{30}{y^4}\).

What does an index of one half mean?

An index of \(\dfrac{1}{2}\) means the square root: \(25^{1/2} = \sqrt{25} = 5\). An index of \(\dfrac{1}{3}\) means the cube root.

How do you simplify a fraction with indices?

Simplify the top and the bottom separately first. Then divide the numbers and subtract the indices of each letter.

What happens when you divide by a negative power?

Subtracting a negative index adds: \(x^5 \div x^{-3} = x^{5+3} = x^8\).

Mixed Index Law Questions

📖 Theory

How to simplify a mixed index question

Common pitfalls

Frequently asked questions

Theory