Exponents & Powers — Class 7 IMO

Laws of Exponents

An exponent tells how many times to multiply a number (the base) by itself.

5 x 5 x 5 = 5^3 (read as '5 to the power of 3' or '5 cubed')

Term	Meaning	Example
Base	The number being multiplied	5
Exponent	How many times to multiply	3
Power	The whole expression	5^3 = 125

Special cases:

a^1 = a (any number to power 1 is itself)
a^0 = 1 (any nonzero number to power 0 is 1)
1^n = 1 (one raised to any power is 1)

EXPONENTIAL GROWTH:
  2^1 = 2    **
  2^2 = 4    ****
  2^3 = 8    ********
  2^4 = 16   ****************

LAWS OF EXPONENTS:
  Same base, multiply: a^m x a^n = a^(m+n)
  Same base, divide:   a^m / a^n = a^(m-n)
  Power of power:      (a^m)^n   = a^(m*n)

Example 1: Simplify 2³ × 2².

2³⁺² = 2⁵ = 32.

Example 2: Simplify (3²)².

3²ˣ² = 3⁴ = 81.

Example 3: Write 7 x 7 x 7 x 7 using exponents.

7^4 (four 7's multiplied together)

Example 4: Write 10^5 as repeated multiplication.

10 x 10 x 10 x 10 x 10 = 100,000

Example 5: Evaluate 2^6.

2 x 2 x 2 x 2 x 2 x 2 = 64

Quick recap

aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ.
a⁰ = 1 for any non-zero a.
Exponent = how many times the base is multiplied by itself
Any nonzero number to power 0 = 1
a^m x a^n = a^(m+n) when bases are the same
(a^m)^n = a^(m*n)

✓ Quick check

Simplify 5² × 5³ as a power of 5.

Add the exponents: 5²⁺³ = 5⁵.

What is 2⁰?

Any non-zero number to the power 0 is 1.

Negative Exponents

Negative bases with exponents: Parentheses make a critical difference!

Expression	Meaning	Result
(-2)^4	(-2) x (-2) x (-2) x (-2)	+16 (positive)
-2^4	-(2 x 2 x 2 x 2)	-16 (negative)

Key rule: Even exponent on a negative base gives a positive result. Odd exponent on a negative base gives a negative result.

Order of Operations (PEMDAS):

Parentheses
Exponents
Multiplication and Division (left to right)
Addition and Subtraction (left to right)

ORDER OF OPERATIONS PYRAMID:

        [Parentheses]    (highest priority)
        [Exponents    ]
        [x or /       ]
        [+ or -       ]  (lowest priority)

NEGATIVE BASE RULE:
  Even exponent: (-3)^4 = +81  (positive)
  Odd exponent:  (-3)^3 = -27  (negative)

Example 1: Find 2⁻³.

1/2³ = 1/8.

Example 2: Find 10⁻².

1/100 = 0.01.

Example 3: Evaluate: (-3)^3 vs -3^3

(-3)^3 = (-3) x (-3) x (-3) = -27. Also -3^3 = -(3x3x3) = -27. Same result because exponent is odd.

Example 4: Evaluate: 3 + 4^2 x 2

Exponents first: 4^2=16. Then multiply: 16 x 2=32. Then add: 3+32=35.

Example 5: Evaluate: (-2)^4 - 3^2

(-2)^4 = 16, 3^2 = 9. Result: 16 - 9 = 7.

Quick recap

a⁻ⁿ = 1/aⁿ.
10⁻¹ = 0.1, 10⁻² = 0.01, and so on.
Even exponent on negative base gives positive result
Odd exponent on negative base gives negative result
Parentheses change the meaning: (-2)^4 is different from -2^4
Always apply PEMDAS: Parentheses, Exponents, Multiply/Divide, Add/Subtract

✓ Quick check

What is 3⁻²?

3⁻² = 1/3² = 1/9.

What is 10⁻¹?

10⁻¹ = 1/10 = 0.1.

Standard Form (Scientific Notation)

Standard form writes a number as a × 10ⁿ, where 1 ≤ a < 10. Large numbers use positive powers (45000 = 4.5 × 10⁴) and small numbers use negative powers (0.0006 = 6 × 10⁻⁴).

Example 1: Write 45000 in standard form.

4.5 × 10⁴.

Example 2: Write 0.0006 in standard form.

6 × 10⁻⁴.

Quick recap

Standard form: a × 10ⁿ with 1 ≤ a < 10.
Large numbers → positive power; small numbers → negative power.

✓ Quick check

Write 32000 in standard form.

32000 = 3.2 × 10⁴.

What is 5 × 10³ as an ordinary number?

5 × 1000 = 5000.