Mixed Index Law Questions
Mixed Index Law Questions combine positive, zero, negative and fractional indices in the same expression. This NSW Year 9 Advanced maths subtopic tests choosing and ordering the index laws (NESA outcome MA5‑IND‑P‑01), with answers written using positive indices.
You will simplify expressions such as \(\frac{(3x^2y)^2}{9xy^3}\), \(\left(\frac{4x^2}{3}\right)^{-2}\) and \(\sqrt[3]{8a^6}\), find the index of one pronumeral in a long product, and work with fractional powers like \(\left(\frac{49}{a^2}\right)^{-\frac{1}{2}}\). You will also find unknown indices in an equation and prove that \(\frac{4^{n+1} \times 8^n}{2^{5n+1}} = 2\) for every whole number \(n\). It is part of Year 9 Advanced Indices and Scientific Notation.
Every question with a fully worked solution.
- Mixed Index Law Questions - Video - Mixed Index Laws 1 Watch
Theory
Mixed index questions use several index laws at once, often with negative, zero and fractional indices. This Year 9 Advanced maths page gives one plan for every mixed question, then applies it to fractional negative powers, single-pronumeral indices and show-that results.
A mixed index question combines the multiplication, division, power-of-a-power and zero index laws in one expression, and may include negative indices, fractional indices or surds.
The laws are the same as always. What makes these questions hard is the order of work and the bookkeeping, so use one fixed plan: clear special indices, expand brackets, collect the top, collect the bottom, divide.
Answers are usually given with positive indices and the coefficient as a whole number or a fraction in simplest form.
A show-that question gives the answer. Your job is the working: every step must follow from a law, and the last line must match what you were asked to show. Substituting one value is a check, not a proof.
| Law | Rule |
|---|---|
| Multiply | \(a^m \times a^n = a^{m+n}\) |
| Divide | \(a^m \div a^n = a^{m-n}\) |
| Power of a power | \((a^m)^n = a^{mn}\) |
| Power of a product or quotient | \((ab)^n\) \(= a^nb^n,\ \left(\dfrac{a}{b}\right)^n\) \(= \dfrac{a^n}{b^n}\) |
| Zero index | \(a^0 = 1\) |
| Negative index | \(a^{-n}\) \(= \dfrac{1}{a^n},\ \left(\dfrac{a}{b}\right)^{-n}\) \(= \left(\dfrac{b}{a}\right)^{n}\) |
| Fractional index | \(a^{m/n} = \left(\sqrt[n]{a}\right)^m\) |
The plan for every mixed question
- Clear special indices: surds become fractional indices; a fraction to a negative power is flipped; anything to the power \(0\) becomes \(1\).
- Expand brackets: raise every factor, multiplying indices. Evaluate number parts by root then power.
- Collect the top and collect the bottom: multiply coefficients, add indices.
- Divide, base by base: subtract indices; simplify the coefficient fraction.
- Positive indices: move any factor with a negative index across the line.
- Check: substitute a small positive value such as \(x = 2\) into the question and the answer.
Variations
- One index only: track a single pronumeral through the laws (Example 3).
- Find unknown indices: simplify in terms of the unknowns, then match indices with the given result.
- Different numerical bases: rewrite \(4, 8, 9, 25, 27, 125\) as prime powers, then the \(n\) terms usually cancel (Example 4).
- Roots of terms: \(\sqrt[3]{64k^9} = (64k^9)^{1/3} = 4k^3\).
Expand the brackets, collect the top and bottom, then divide.
| \((2a^3b^{-1})^3\) | \(=\) | \(8a^9b^{-3}\) |
| \((6a^2b)^2\) | \(=\) | \(36a^4b^2\) |
| \(\text{top}\) | \(=\) | \(8a^9b^{-3} \times 3a^{-2}b^5 = 24a^{7}b^{2}\) |
| \(\dfrac{24a^7b^2}{36a^4b^2}\) | \(=\) | \(\dfrac{2}{3}a^{7-4}b^{2-2}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{2a^3}{3}}\) |
\(b^{-3} \times b^5 = b^2\), which cancels with \(b^2\) underneath: \(b^0 = 1\).
Flip the fraction to clear the negative, then take the square root of each factor.
| \(\left(\dfrac{9x^4}{y^6}\right)^{-1/2}\) | \(=\) | \(\left(\dfrac{y^6}{9x^4}\right)^{1/2}\) |
| \(\) | \(=\) | \(\dfrac{y^3}{3x^2}\) |
| \(\dfrac{y^3}{3x^2} \times \dfrac{6x^3}{y^2}\) | \(=\) | \(\dfrac{6x^3y^3}{3x^2y^2}\) |
| \(\) | \(=\) | \(\boldsymbol{2xy}\) |
\(\sqrt{9} = 3\), \((x^4)^{1/2} = x^2\) and \((y^6)^{1/2} = y^3\).
Only \(b\) matters. Find its index in each part, then add or subtract.
| \((a^2bc^3)^5\) | \(\rightarrow\) | \(b^{1 \times 5} = b^5\) |
| \((b^4c)^3\) | \(\rightarrow\) | \(b^{4 \times 3} = b^{12}\) |
| \((abc)^6\) | \(\rightarrow\) | \(b^{6}\) |
| \(b^5 \times b^{12} \div b^6\) | \(=\) | \(b^{5+12-6} = b^{11}\) |
The index of \(b\) is \(\mathbf{11}\).
A \(b\) with no index written still counts as \(b^1\), and the outside index multiplies it.
Write every base as a power of \(5\).
| \(25^n\) | \(=\) | \((5^2)^n = 5^{2n}\) |
| \(125^n\) | \(=\) | \((5^3)^n = 5^{3n}\) |
| \(\dfrac{5^{2n} \times 5^{n+2}}{5^{3n}}\) | \(=\) | \(5^{2n + n + 2 - 3n}\) |
| \(\) | \(=\) | \(5^{2} = \mathbf{25}\) |
The \(n\) terms cancel in the index, so the value does not depend on \(n\). Testing one value of \(n\) is a check, not a proof.
Common pitfalls
Frequently asked questions
What order should you apply the index laws in?
Clear any surds, negative outer indices and zero indices first, then expand brackets, collect the top and the bottom, and divide base by base. Finish with positive indices.
How do you simplify an expression with negative and fractional indices together?
If a fraction has a negative fractional power, flip it to make the index positive, then take the root and power of each factor. After that use the ordinary laws.
How do you find the index of just one pronumeral?
Follow only that pronumeral: multiply its index by any outside power, add for multiplication and subtract for division. The other pronumerals can be ignored.
How do you show an index expression is the same for every n?
Write each base as a power of the same prime, combine the indices with the laws, and show that the terms in \(n\) cancel to leave a number.
Is substituting a number enough to prove an index identity?
No. One value can only check the result. A proof uses the index laws on the general expression so that it works for every value.