Mastering Indices: Powers, Roots, and Reciprocals

1. What are Indices?

An index, or power, tells you how many times to multiply a number by itself. For example, in the expression $5^3$, the number 5 is the base and the number 3 is the index (or power). It means "multiply 5 by itself 3 times":

$5^3=5\times 5\times 5=125$

Given your IT background, you've likely encountered this concept frequently, especially with powers of 2, which are fundamental to computing (e.g., $2^{10}=1024$, which is the basis for measurements like kilobytes and megabytes).

To get a formal introduction, please watch the first minute of this video.

2. The Basic Rules of Indices

To work with indices efficiently, we use a set of rules or laws. We'll start with the three most fundamental ones, which apply to integer indices. For a detailed explanation and derivation of these rules, we'll turn to a well-structured guide.

Here is a summary of the rules you just read about:

1. Multiplication Rule: When multiplying powers with the same base, you add the indices.
   $a^m\times a^n=a^{m+n}$
   Example: $x^3\times x^4=x^{3+4}=x^7$

2. Division Rule: When dividing powers with the same base, you subtract the indices.
   $a^m÷a^n=a^{m-n}$
   Example: $y^8÷y^5=y^{8-5}=y^3$

3. Power of a Power Rule: When raising a power to another power, you multiply the indices.
   $(a^m{)}^n=a^{mn}$
   Example: $(b^2{)}^5=b^{2\times 5}=b^{10}$

3. The Zero Index and Negative Indices

The rules of indices lead to some interesting and important special cases. What does it mean for a number to have an index of 0 or a negative number?

Let's return to the Indices or Powers guide to see how these are derived from the division rule.

As you saw, these are not arbitrary rules but logical consequences of the division rule:

4. Zero Index Rule: Any non-zero base raised to the power of 0 is 1.
   $a^0=1(\text{for }a\ne 0)$
   This comes from the fact that, for example, $a^3÷a^3=a^{3-3}=a^0$, and we also know that any number divided by itself is 1.

5. Negative Index Rule: A negative index means you take the reciprocal of the base raised to the positive index.
   $a^{-m}=\frac{1}{a^m}$
   This comes from expressions like $a^2÷a^5=a^{2-5}=a^{-3}$. By cancelling terms, we also see that $\frac{a\times a}{a\times a\times a\times a\times a}=\frac{1}{a^3}$. Therefore, $a^{-3}=\frac{1}{a^3}$.

4. Fractional Indices

We've covered integer indices. Now, what happens if the index is a fraction, like $\frac{1}{2}$ or $\frac{2}{3}$?

• A fractional index indicates a root. The denominator of the fraction tells you which root to take.
• The numerator indicates a power.

So, for an expression like $a^{p/q}$:

• The denominator $q$ means take the $q$-th root of $a$.
• The numerator $p$ means raise the result to the power of $p$.

$a^{p/q}=(\sqrt[q]{a}{)}^p\text{or}\sqrt[q]{a^p}$

It's usually much easier to take the root first, then apply the power, as this keeps the numbers smaller.

This concept can be tricky, so we'll watch a video that breaks it down very clearly.

The same rules apply when the base itself is a fraction.

Let's continue with the same video to see this in action.

Conclusion

In this lesson, you have learned the essential properties for manipulating expressions with indices. These rules are fundamental tools that you will use constantly throughout algebra and calculus.

Key Takeaways:

• Indices provide a compact notation for repeated multiplication.
• The laws of indices allow for the simplification of complex expressions involving multiplication, division, and powers.
• A zero index results in 1 ($a^0=1$).
• A negative index signifies a reciprocal ($a^{-m}=1/a^m$).
• A fractional index represents a root and a power ($a^{p/q}=(\sqrt[q]{a}{)}^p$).

To help you remember these rules, here is a handy summary table.

Preview of the Next Lesson:

Now that we have a solid understanding of indices, we are ready to tackle our first major topic in this module: quadratic expressions. A quadratic expression is a polynomial of degree 2, like $a{x}^2+bx+c$. In the next lesson, we will learn how to factorise (factor) these expressions, which is a key technique for solving quadratic equations.