Strand 3 — Number

Indices: Laws, Negative and Fractional Powers

3rd Year (Junior Cert)

✓By the end of this lesson students will be able to understand and apply the fundamental laws of indices.
✓By the end of this lesson students will be able to evaluate expressions involving a zero index.
✓By the end of this lesson students will be able to evaluate expressions involving negative indices.
✓By the end of this lesson students will be able to evaluate expressions involving fractional indices, converting between index and root form.
✓By the end of this lesson students will be able to simplify complex expressions using a combination of index laws.

Key concepts

What is an Index?

An index (also called a power or exponent) tells us how many times a number (the base) is multiplied by itself. For example, in 2^3, 2 is the base and 3 is the index. It means 2 × 2 × 2.

a^n = a × a × ... × a (n times)

Law 1: Multiplication of Indices

When multiplying terms with the same base, we add their indices.

a^m × a^n = a^(m+n)

Law 2: Division of Indices

When dividing terms with the same base, we subtract their indices.

a^m ÷ a^n = a^(m-n)

Law 3: Power of a Power

When raising a power to another power, we multiply the indices.

(a^m)^n = a^(mn)

Law 4: Power of a Product

When a product is raised to a power, each factor in the product is raised to that power.

(ab)^n = a^n × b^n

Law 5: Power of a Quotient

When a quotient (fraction) is raised to a power, both the numerator and the denominator are raised to that power.

(a/b)^n = a^n / b^n

Zero Index

Any non-zero number raised to the power of zero is equal to 1.

a^0 = 1 (where a ≠ 0)

Negative Indices

A negative index indicates the reciprocal of the base raised to the positive index. It means 1 divided by the base raised to the positive power.

a^(-n) = 1 / a^n

Fractional Indices

A fractional index represents a root. For example, a power of 1/2 means the square root, and a power of 1/3 means the cube root. More generally, a^(m/n) means the nth root of a, raised to the power of m.

a^(1/n) = nth√a; a^(m/n) = (nth√a)^m = nth√(a^m)

Key facts to remember

1a^m × a^n = a^(m+n)
2a^m ÷ a^n = a^(m-n)
3(a^m)^n = a^(mn)
4a^0 = 1 (for a ≠ 0)
5a^(-n) = 1 / a^n
6a^(1/n) = nth√a
7a^(m/n) = (nth√a)^m = nth√(a^m)
8(ab)^n = a^n b^n and (a/b)^n = a^n / b^n

Worked examples

Example 1

Simplify the following expression: (3x^2y^3)^2 × (2x^4y)^-1

IApply Law 4 and Law 3 to the first term: (3x^2y^3)^2 = 3^2 × (x^2)^2 × (y^3)^2 = 9x^(2×2)y^(3×2) = 9x^4y^6.

IIApply the negative index rule to the second term: (2x^4y)^-1 = 1 / (2x^4y)^1 = 1 / (2x^4y).

IIIMultiply the simplified terms: (9x^4y^6) × (1 / (2x^4y)).

IVCombine the terms: (9x^4y^6) / (2x^4y).

VApply Law 2 for division of indices (x^4/x^4 = x^(4-4) = x^0 = 1; y^6/y^1 = y^(6-1) = y^5).

VISimplify the numerical coefficients: 9/2.

VIICombine all simplified parts.

Answer

(9y^5)/2

Remember to apply the power to all parts of the product inside the bracket, including the numerical coefficient.

Example 2

Evaluate: (81)^(3/4)

IRecognise that the fractional index 3/4 means the 4th root of 81, raised to the power of 3. (81)^(3/4) = (4th√81)^3.

IICalculate the 4th root of 81. We need a number that, when multiplied by itself 4 times, gives 81. We know 3 × 3 × 3 × 3 = 81, so 4th√81 = 3.

IIIRaise the result to the power of 3: 3^3.

IVCalculate 3^3 = 3 × 3 × 3.

Answer

It's often easier to find the root first, then apply the power, especially with larger numbers.

Example 3

Simplify: (a^5b^-2) / (a^-3b^4)

ISeparate the terms with the same base: (a^5 / a^-3) × (b^-2 / b^4).

IIApply Law 2 (division of indices) for base 'a': a^5 / a^-3 = a^(5 - (-3)) = a^(5+3) = a^8.

IIIApply Law 2 (division of indices) for base 'b': b^-2 / b^4 = b^(-2 - 4) = b^-6.

IVCombine the simplified terms: a^8 × b^-6.

VConvert the negative index to a positive index using the negative index rule: b^-6 = 1 / b^6.

VIWrite the final expression with positive indices.

Answer

a^8 / b^6

Be careful with subtracting negative numbers when applying the division law.

Common mistakes

✗Adding indices when multiplying different bases (e.g., x^2 × y^3 ≠ (xy)^5).
✗Forgetting that a^0 = 1, often mistakenly writing a^0 = 0.
✗Incorrectly handling negative indices, for example, thinking a^-n = -a^n or a^-n = 1/(-a^n).
✗Applying the power only to the variable and not the coefficient (e.g., (2x)^3 ≠ 2x^3, it should be 8x^3).
✗Confusing the order of operations with fractional indices, e.g., (x^1/2)^3 vs x^(1/2 × 3).

Exam tips

★Always show your working step-by-step, especially when applying multiple index laws. This helps in identifying errors and earns partial credit.
★Simplify expressions fully, ensuring all final answers have positive indices unless otherwise specified.
★When dealing with fractional indices, remember that the denominator indicates the root and the numerator indicates the power. You can often choose to find the root first or apply the power first, but finding the root first is usually easier for numerical problems.
★Practice converting between index form and root form, as this is a common requirement in exam questions.

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