The Laws of Indices (Junior Cert Mathematics): Revision Notes

The Laws of Indices

An exponent, also known as a power, tells you how many times to multiply a number by itself. For example, in the expression $(2^3),$ the number $2$ is the base, and the number $3$ is the exponent or power. This means $2^3$ is equal to $2 \times 2 \times 2 = 8$.

In simpler terms, the power tells you how many times to use the base number in a multiplication. So, whenever you see an exponent or power, think of it as repeated multiplication.

When dealing with expressions containing indices we may use the law of indices to simplify them.

The Laws of Indices (Rules for Working with Powers)

1. Product of Powers
   you had a s $a^p \times a^q = a^{p+q}$ Explanation: When multiplying two numbers with the same base, add the powers.
2. Quotient of Powers: $\frac{a^p}{a^q} = a^{p-q}, \quad a \neq 0$ Explanation: When dividing two numbers with the same base, subtract the power in the denominator from the power in the numerator.
3. Power of a Power: $\left(a^p\right)^q = a^{p \times q}$ Explanation: When raising a power to another power, multiply the powers together.
4. Zero Power: $a^0 = 1, \quad a \neq 0$ Explanation: Any number raised to the power of zero equals 1.
5. Negative Power: $a^{-p} = \frac{1}{a^p}, \quad a \neq 0$ Explanation: A negative power means you take the reciprocal (flip) of the number and then apply the positive power.
6. Fractional Power: $a^{\frac{1}{q}} = \sqrt[q]{a}$ Explanation: A fractional power indicates a root. For example, $a^{\frac{1}{2}}$ is the square root of $a$, and $a^{\frac{1}{3}}$ is the cube root of $a$.
7. General Fractional Power: $a^{\frac{p}{q}} = \sqrt[q]{a^p} = \left(\sqrt[q]{a}\right)^p$ Explanation: A general fractional power involves both a root and a power. You first take the root, then raise it to the power.
8. Power of a Product: $(ab)^p = a^p \times b^p$ Explanation: When raising a product (two numbers multiplied together) to a power, raise each number to that power separately.
9. Power of a Quotient: $\left(\frac{a}{b}\right)^p = \frac{a^p}{b^p}, \quad b \neq 0$ Explanation: When raising a fraction to a power, raise both the numerator (top number) and the denominator (bottom number) to that power separately.

Important Points to Remember:

10. Keep the Base the Same: The laws of indices only work when the base is the same. Always check that the base number hasn't changed before applying the rules.
11. Negative Powers: Don't be confused by negative powers. They don't make the number negative—they just mean you take the reciprocal of the base and then apply the positive power.
12. Zero Power Rule: Remember that any number raised to the power of zero equals 1. This is a special rule that often appears in problems.
13. Understanding Roots and Fractional Powers: Get comfortable with the idea that fractional powers are related to roots. For example, $a^{\frac{1}{2}}$is just another way of writing the square root of $a$.
14. Combining Laws: Sometimes, you'll need to use more than one of these laws to simplify an expression. Take it step by step, using the appropriate rule for each part of the problem.