Indices

 

What Are Indices (Powers)?

Indices, also known as powers or exponents, are a way of showing repeated multiplication.

Instead of writing out $2 \times 2 \times 2$, you can use an index (or power) to write it more simply $2^3$

• Base: The number being multiplied, like the $2$ in $2^3$
• Exponent/Power: How many times the base is multiplied by itself, like the $3$ in $2^3$

So, $2^3 = 2 \times 2 \times 2 = 8$

 

Power Laws (Rules of Indices)

When you’re working with powers, there are some handy rules to remember. These make calculations easier when you’re multiplying, dividing, or raising powers.

 

1. Multiplying Powers (Same Base)

When you multiply numbers with the same base, you add the powers.

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

Example: $$2^3 \times 2^4 = 2^{3+4} = 2^7$$

 

2. Dividing Powers (Same Base)

When you divide numbers with the same base, you subtract the powers.

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

Example: $$5^6 \div 5^2 = 5^{6-2} = 5^4$$

 

3. Power of a Power

When you raise a power to another power, you multiply the exponents.

$$(a^m)^n = a^{m \times n}$$

Example: $$(3^2)^4 = 3^{2 \times 4} = 3^8$$

 

4. Power of 1

Any number to the power of $1$ is itself 

$$a^1 = a   i.e   7^1 = 7$$

5. Power of 0

Any number to the power of $0$ is $1$

$$a^0 = 1$$

Example: $$9^0 = 1$$

 

6. Negative Powers

A negative power flips the number to a fraction (reciprocal).

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

Example: $$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$$

 

7. Single Fractional Powers:

• When we see $a^{\frac{1}{n}}$, , it’s the same as saying “the n-th root of aaa”:
• So, $a^{\frac{1}{n}} = \sqrt[n]{a}$

Example: $$16^{\frac{1}{4}} = \sqrt[4]{16} = 2$$

 

8. Fractional Powers:

• A fractional power, like $a^{\frac{m}{n}}$, means we’re dealing with both a root and a power.
• Think of it like this: $a^{\frac{m}{n}} = \left( \sqrt[n]{a} \right)^m = \sqrt[n]{a^m}$
• You can take the root first, then apply the power, or apply the power first, then the root. Either way, you'll end up with the same answer.

Example: $$27^{\frac{2}{3}} =  \left( \sqrt[3]{27} \right)^2 = 3^2 = 9$$

 

9. Negative Fractional Powers:

• Negative fractional powers combine both concepts: the root and the reciprocal (flipping).
• So, $a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{\left( \sqrt[n]{a} \right)^m}$

Example: $$8^{-\frac{2}{3}} = \frac{1}{8^{\frac{2}{3}}} = \frac{1}{4}$$

Breaking it down:

• First, calculate $8^{\frac{2}{3}} = 4$ (since $\sqrt[3]{8} = 2$ and $2^2 = 4$ ).
• Then flip it: $8^{-\frac{2}{3}} = \frac{1}{4}$