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.

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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.

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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.

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