The Laws of Indices Revision Notes for Leaving Cert Mathematics

The Laws of Indices

What are indices?

Indices (also called powers or exponents) tell us how many times a number is multiplied by itself. In the expression $2^3$ , the number 2 is called the base and 3 is the index or power.

For example: $2^3 = 2 × 2 × 2 = 8$

We say this as "2 cubed" or "2 to the power of 3". The index 3 tells us that the number 2 is multiplied by itself 3 times.

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Understanding the terminology is crucial for working with indices. The base is the number being multiplied, while the index (or power) tells you how many times to multiply the base by itself.

The fundamental laws of indices

Law 1: Multiplication of powers with the same base

When you multiply powers that have the same base, you add the indices.

Formula: $a^m × a^n = a^{m+n}$

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Worked Example: Multiplying Powers

$4^2 × 4^3 = (4 × 4) × (4 × 4 × 4) = 4^5$
$x^2 × x^3 = x^{2+3} = x^5$

Law 2: Division of powers with the same base

When you divide powers that have the same base, you subtract the indices.

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

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Worked Example: Dividing Powers

$\frac{3^5}{3^2} = \frac{3×3×3×3×3}{3×3} = 3^3$
$\frac{x^5}{x^2} = x^{5-2} = x^3$

Law 3: Power to a power

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

Formula: $(a^m)^n = a^{m×n}$

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Worked Example: Power to a Power

$(x^2)^3 = x^2 × x^2 × x^2 = x^{2+2+2} = x^6 = x^{2×3}$
$(x^4)^3 = x^{4×3} = x^{12}$

Law 4: Product raised to a power

When you raise a product to a power, each factor gets raised to that power.

Formula: $(ab)^n = a^n × b^n$

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This law is particularly useful when dealing with expressions involving multiple variables or numbers multiplied together.

Law 5: Zero power rule

Any number (except zero) raised to the power of zero equals 1.

Formula: $a^0 = 1$

Examples: $5^0 = 1$ , $10^0 = 1$ , $x^0 = 1$

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Remember that $0^0$ is undefined in mathematics. This rule only applies to non-zero bases.

Law 6: Negative indices

A negative index means you take the reciprocal and make the index positive.

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

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Worked Example: Negative Indices

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

You can also think of it as: $\frac{4^3}{4^5} = 4^{3-5} = 4^{-2} = \frac{1}{4^2}$

Law 7: Fractional indices

Fractional indices are connected to roots. A fractional index of $\frac{1}{2}$ means square root.

Formula: $a^{\frac{1}{n}} = \sqrt[n]{a}$

Key relationships:

$2^{\frac{1}{2}} = \sqrt{2}$
$2^{\frac{1}{3}} = \sqrt[3]{2}$
$a^{\frac{1}{n}} = \sqrt[n]{a}$

We can prove this using the multiplication rule: $2^{\frac{1}{2}} × 2^{\frac{1}{2}} = 2^{\frac{1}{2}+\frac{1}{2}} = 2^1 = 2$

Since $\sqrt{2} × \sqrt{2} = 2$ , we know that $2^{\frac{1}{2}} = \sqrt{2}$ .

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This connection between fractional indices and roots is fundamental. The denominator of the fraction tells you which root to take, while the numerator (when it's not 1) tells you what power to raise the result to.

Law 8: More complex fractional indices

For fractional indices like $\frac{m}{n}$ , we use the rule $(a^m)^n = a^{mn}$ .

Formula: $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$

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Worked Example: Complex Fractional Indices

$8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$
$16^{\frac{3}{4}} = (\sqrt[4]{16})^3 = 2^3 = 8$

Worked examples

Example 1: Simplifying expressions

Write each of these as whole numbers or fractions:

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Worked Example: Simplifying Complex Expressions

(i) $\frac{3^4 × 3^2}{3^5}$

Solution: $\frac{3^4 × 3^2}{3^5} = \frac{3^{4+2}}{3^5} = \frac{3^6}{3^5} = 3^{6-5} = 3^1 = 3$

(ii) $64^{\frac{1}{3}}$

Solution: $64^{\frac{1}{3}} = \sqrt[3]{64} = 4$ (since $4^3 = 64$ )

(iii) $\frac{1}{4^{-2}}$

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

(iv) $8^{\frac{2}{3}}$

Solution: $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$

Example 2: Converting to standard form

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Worked Example: Standard Form Conversions

(i) Express $\left(\frac{8}{27}\right)^{\frac{2}{3}}$ in the form $\frac{a}{b}$ , where $a, b ∈ ℕ$ .

Solution: $\left(\frac{8}{27}\right)^{\frac{2}{3}} = \frac{8^{\frac{2}{3}}}{27^{\frac{2}{3}}} = \frac{(\sqrt[3]{8})^2}{(\sqrt[3]{27})^2} = \frac{2^2}{3^2} = \frac{4}{9}$

(ii) Express $\frac{125}{\sqrt{5}}$ as a power of 5.

Solution: $\frac{125}{\sqrt{5}} = \frac{5^3}{5^{\frac{1}{2}}} = 5^{3-\frac{1}{2}} = 5^{\frac{5}{2}}$

Example 3: Working with variables

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Worked Example: Variable Expressions

Simplify $\frac{x^7}{x^3} × (x^2)^4$

Solution: $\frac{x^7}{x^3} × (x^2)^4 = x^{7-3} × x^{2×4} = x^4 × x^8 = x^{4+8} = x^{12}$

Common exam tips

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Essential Exam Strategies:

Always check your base numbers - the laws only apply when the bases are the same
Remember the order of operations - brackets first, then indices, then multiplication/division
Convert roots to fractional indices when working with mixed expressions
Watch out for negative indices - they create fractions, not negative numbers
Zero to any power is zero, but any number to the power zero is one

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Common mistakes to avoid:

Forgetting that the bases must be the same before applying the laws
Confusing negative indices with negative numbers
Mixing up the rules for multiplication and division of indices

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Key Points to Remember:

Multiplication of same bases: Add the indices ( $a^m × a^n = a^{m+n}$ )
Division of same bases: Subtract the indices ( $\frac{a^m}{a^n} = a^{m-n}$ )
Power to a power: Multiply the indices ( $(a^m)^n = a^{mn}$ )
Negative indices create reciprocals: $a^{-n} = \frac{1}{a^n}$
Fractional indices are roots: $a^{\frac{1}{n}} = \sqrt[n]{a}$

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