Section A — Chapters 1–22

Indices and Scientific Notation

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to understand and use index notation.
✓By the end of this lesson students will be able to calculate square roots of perfect squares.
✓By the end of this lesson students will be able to apply the first, second, and third laws of indices.
✓By the end of this lesson students will be able to express numbers in scientific notation and convert them back to ordinary numbers.
✓By the end of this lesson students will be able to perform basic calculations involving indices and scientific notation.

Key concepts

Index (Power) and Base

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

a^n = a × a × ... × a (n times)

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. The symbol for square root is √. For example, the square root of 25 is 5 because 5 × 5 = 25.

√a = b if b × b = a

First Law of Indices (Multiplication)

When multiplying powers with the same base, you add the indices (powers).

a^m × a^n = a^(m+n)

Second Law of Indices (Division)

When dividing powers with the same base, you subtract the indices (powers).

a^m ÷ a^n = a^(m-n)

Third Law of Indices (Power of a Power)

When raising a power to another power, you multiply the indices (powers).

(a^m)^n = a^(m×n)

Scientific Notation

Scientific notation is a way of writing very large or very small numbers using powers of 10. It is written in the form a × 10^n, where 'a' is a number between 1 and 10 (but not including 10), and 'n' is an integer. If 'n' is positive, it's a large number; if 'n' is negative, it's a small number.

a × 10^n (where 1 ≤ |a| < 10 and n is an integer)

Key facts to remember

1An index (or power) tells you how many times to multiply the base number by itself (e.g., 5^3 = 5 × 5 × 5).
2The square root of a number 'x' is a number 'y' such that y × y = x (e.g., √49 = 7 because 7 × 7 = 49).
3First Law of Indices: When multiplying powers with the same base, add the indices (a^m × a^n = a^(m+n)).
4Second Law of Indices: When dividing powers with the same base, subtract the indices (a^m ÷ a^n = a^(m-n)).
5Third Law of Indices: When raising a power to another power, multiply the indices ((a^m)^n = a^(m×n)).
6Scientific notation is written as a × 10^n, where 'a' is a number between 1 and 10 (1 ≤ a < 10) and 'n' is an integer.
7A positive 'n' in scientific notation indicates a large number, while a negative 'n' indicates a small number.

Worked examples

Example 1

Calculate the value of 4^3 and find the square root of 100.

IFor 4^3: This means 4 multiplied by itself 3 times.

II4^3 = 4 × 4 × 4

III4 × 4 = 16

IV16 × 4 = 64

VFor √100: We need to find a number that, when multiplied by itself, gives 100.

VIWe know that 10 × 10 = 100.

Answer

4^3 = 64, √100 = 10

Remember to multiply the base by itself the number of times indicated by the index, not multiply the base by the index.

Example 2

Simplify the expression: (3^2 × 3^4) ÷ 3^3

IFirst, apply the First Law of Indices for the multiplication part: 3^2 × 3^4 = 3^(2+4) = 3^6.

IINow the expression becomes: 3^6 ÷ 3^3.

IIINext, apply the Second Law of Indices for the division part: 3^6 ÷ 3^3 = 3^(6-3) = 3^3.

IVFinally, calculate the value: 3^3 = 3 × 3 × 3 = 27.

Answer

Always work from left to right, following the order of operations. Simplify inside brackets first if present.

Example 3

a) Write 78,000,000 in scientific notation. b) Convert 5.1 × 10^-4 to an ordinary number.

Ia) To write 78,000,000 in scientific notation:

IIMove the decimal point to the left until there is only one non-zero digit before it. The number becomes 7.8.

IIICount how many places the decimal point moved. It moved 7 places to the left.

IVSince the original number is large, the power of 10 is positive.

VSo, 78,000,000 = 7.8 × 10^7.

VIb) To convert 5.1 × 10^-4 to an ordinary number:

VIIThe power of 10 is -4, which means the number is small, and we need to move the decimal point to the left.

VIIIMove the decimal point 4 places to the left from its current position in 5.1.

95.1 → 0.51 → 0.051 → 0.0051 → 0.00051.

Answer

a) 7.8 × 10^7, b) 0.00051

A positive power of 10 means a large number (move decimal right). A negative power of 10 means a small number (move decimal left).

Common mistakes

✗Multiplying the base by the index instead of repeated multiplication (e.g., calculating 2^3 as 2 × 3 = 6 instead of 2 × 2 × 2 = 8).
✗Applying the laws of indices when the bases are different (e.g., trying to simplify 2^3 × 3^2 using the first law).
✗Making errors when counting decimal places or moving the decimal point for scientific notation, especially with negative powers.
✗Not ensuring the 'a' part of scientific notation (a × 10^n) is between 1 and 10 (e.g., writing 25 × 10^3 instead of 2.5 × 10^4).
✗Incorrectly calculating square roots (e.g., dividing the number by 2 instead of finding the number that squares to it).

Exam tips

★Always show all your steps clearly when applying the laws of indices. This helps you get partial marks even if your final answer is incorrect.
★Remember the order of operations (BODMAS/PEMDAS) when simplifying expressions involving indices.
★For scientific notation, double-check the direction and number of places you move the decimal point, and ensure 'a' is in the correct range.
★Practise each law of indices separately before attempting problems that combine them.

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