C2.4 Indices II

Learning objectives

C2.4.1Understand and use indices (positive, zero and negative).• Understand and use indices (positive, zero and negative).C2.4.2Understand and use the rules of indices• Understand and use the rules of indices. e.g. 2x = 32. Find the value of x. e.g. simplify: • (5x 3) 2 • 12a 5 ÷ 3a –2 • 6x 7y 4 × 5x –5y. Knowledge of logarithms is not required.

In $a^n$ , $a$ is the base and $n$ is the index. The index describes repeated multiplication when it is positive, and the same pattern extends consistently to zero and negative indices.

Index type	Meaning	Example
positive	multiply the base by itself	$3^4=3 imes3 imes3 imes3=81$
zero	the value is 1 for a non-zero base	$7x^0=7 imes1=7$
negative	take the reciprocal of the matching positive power	$x^{-3}=1/x^3$ for $x
e0$

Each step down in the index divides by the base: $5^3=125$ , $5^2=25$ , $5^1=5$ , $5^0=1$ , $5^{-1}=1/5$ . This pattern explains both the zero-index and negative-index meanings.

The index applies only to its stated base. In $(-2)^4$ , the base is $-2$ , so the value is 16. In $-2^4$ , the power applies to 2 first, so the value is $-16$ .

A negative index does not make the value negative: $y^{-2}=1/y^2$ . It moves a non-zero factor across a fraction bar and changes the sign of the index.

$a^0=1$ and $a^{-n}=1/a^n$ require $a
e0 $; division by zero is undefined. Do not multiply the base by the index:$ 4^3 $means$ 4 imes4 imes4 $, not$ 4 imes3$.

Index laws preserve the meaning of repeated multiplication. First identify the operation and confirm that the bases match; then change the indices using the corresponding rule.

Structure	Rule	Example
same base, multiply	$a^m a^n=a^{m+n}$	$x^3x^5=x^8$
same base, divide	$a^m/a^n=a^{m-n}$	$y^3/y^5=y^{-2}=1/y^2$
power of a power	$(a^m)^n=a^{mn}$	$(w^5)^4=w^{20}$
power of a product	$(ab)^n=a^nb^n$	$(3x^2y^4)^3=27x^6y^{12}$

Treat numerical coefficients separately from variable powers. For $6x^7y^4 imes5x^{-5}y$ , multiply coefficients and add indices for each matching base: $30x^{7+(-5)}y^{4+1}=30x^2y^5$ .

For $12a^5\div3a^{-2}$ , divide coefficients and subtract the denominator index: $4a^{5-(-2)}=4a^7$ . Subtracting a negative index increases the result's index.

To find an unknown index, write both sides with the same base and equate indices. Since $2^x=32=2^5$ , $x=5$ . Likewise, $9^{12}/9^w=9^4$ gives $12-w=4$ , so $w=8$ ; logarithms are not required.

Add or subtract indices only when multiplying or dividing powers with the same base. There is no rule that turns $a^m+a^n$ into $a^{m+n}$ , and $(a^m)^n$ uses multiplication of indices, not addition.

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C2.4 Indices II

Learning objectives

Interpret positive, zero and negative indices

Choose and apply the index laws

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