Powers of powers and zero indices
For positive integer inner and outer indices, count repeated groups by multiplication. For index zero, the base must be nonzero. Brackets determine whether a coefficient is inside the powered base
\[\begin{gathered}(a^m)^n=a^{mn}\\a^0=1\quad(a\ne0)\end{gathered}\]
- Multiply the two indices For positive integer indices, the outer index counts copies of the inner group \(\displaystyle (a^3)^2=a^{3\times2}=a^6\)
- Brackets control the scope An outside coefficient is not raised. A coefficient inside the powered brackets is raised \(\displaystyle \begin{gathered}4(x^2)^3=4x^6\\(4x^2)^3=64x^6\end{gathered}\)
- Zero power requires a nonzero base The quotient of a nonzero power by itself is 1. The zero-index rule expresses this cancellation \(\displaystyle \begin{gathered}\frac{a^3}{a^3}=a^{3-3}\\=a^0=1\quad(a\ne0)\end{gathered}\)
- Distinguish a powered product from an outside factor For x ≠ 0, the zero power affects only the expression inside its brackets \(\displaystyle \begin{gathered}(5x)^0=1\\5x^0=5\end{gathered}\)
Worked example
Simplify the expression using index law 3.
\[3\left(y^{2}\right)^{3}\]
- \(\displaystyle 3\left(y^{2}\right)^{3}=3y^{2\times3}\)The coefficient 3 is outside the brackets, and the outer index applies to y.
- \(\displaystyle 2\times3=6\)Multiply the inner and outer indices
- \(\displaystyle 3y^{2\times3}=3y^{6}\)Replace the index product with 6 and keep the outside coefficient 3.
Common mistakes
Adding indices for a power of a power
✕\(\displaystyle \left(x^{3}\right)^{2}=x^{3+2}=x^{5}\)✓\(\displaystyle \left(x^{3}\right)^{2}=x^{3\times2}=x^{6}\)
A power raised to a power multiplies the inner and outer indicesRaising an outside coefficient
✕\(\displaystyle 4\left(x^{2}\right)^{3}=4^{3}x^{6}=64x^{6}\)✓\(\displaystyle 4\left(x^{2}\right)^{3}=4x^{6}\)
The coefficient 4 is outside the brackets, so the outer power does not apply to itApplying zero power to a coefficient
✕\(\displaystyle 6x^{0}=\left(6x\right)^{0}=1\)✓\(\displaystyle 6x^{0}=6\times1=6\)
The zero power is attached to x only, so the coefficient 6 remains unchanged. This assumes x is nonzeroDifficulty levels
introSimplify a single power of a power by multiplying its inner and outer indices
standardApply index law 3 to a power of a pronumeral, including an outside coefficient
challengeCombine powers of powers or a zero-power term inside a larger expression
examRecognise the correct nested-power form while preserving an outside coefficient
Questions about this topic
Why are the indices multiplied in a power of a power?+
The inner power repeats the base, and the outer power repeats that group, so the total number of base factors is the product of the two indices
Does the outer power apply to a coefficient outside the brackets?+
No. The coefficient 4 is outside the powered brackets, so it stays 4.
Why is a nonzero base to index 0 equal to 1?+
The quotient of a nonzero power by the same power is 1, and the quotient law identifies this with the zero power
Can I apply the zero-index rule to base 0?+
No. The rule stated here requires a nonzero base. Its cancellation proof cannot divide by zero
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 6: Indices and surds
In this chapter
Work with index notationExpress a number in prime-factor index formSimplify using index laws 1 and 2Simplify using index law 3 and the zero powerExpand using index laws 4 and 5Work with negative indicesSolve an exponential equation by matching basesCombine positive and negative integer indicesScientific notationMultiply and divide in scientific notationSI prefixes and scientific timescalesRound to decimal places and significant figuresRecord measurements and compare instrumentsRelate fractional indices to roots (optional extension)Locate exact square roots on the real number lineSimplify, compare and operate with surds (optional practice)