Important Formulas Exponents and Powers - (Maths) Class 7 (Old NCERT) PDF

Important Formulas

(1) If a is a non‐zero rational number and n is a natural number, then the product a × a × a × ... × a (n times) is denoted by aⁿ and is read as 'a raised to the power n'. The rational number a is called the base and the natural number n is called the exponent. The form aⁿ is also called the exponential form of the repeated product a × a × ... × a.

(2) Special exponents:

• For any non‐zero rational number a, a⁰ = 1.
• For any rational number a, a¹ = a.
• 0⁰ is undefined; avoid using it.

(3) Laws of exponents - If a and b are non‐zero rational numbers and m, n are natural numbers, then the following rules hold:

Explanations and Simple Proofs

a^m × aⁿ = a^m+n

Explanation: The left side represents a multiplied by itself m times and then multiplied by itself n more times. Together there are m + n factors of a, so the result is a^m+n.

a^m ÷ aⁿ = a^m−n (for a = 0)

Explanation: Division cancels common factors. If m ≥ n, cancelling n factors from numerator leaves m − n factors of a. If m < n, cancellation leaves factors in denominator; this leads to negative exponents (introduced later), but for natural number exponents the formula expresses cancellation.

(a^m)ⁿ = a^mn

Explanation: Raising a^m to the power n means multiplying a^m by itself n times. Each factor contributes m copies of a, so total copies = m × n.

(ab)ⁿ = aⁿ bⁿ

Explanation: (ab)(ab)...(ab) (n times) expands to a×a×...×a multiplied by b×b×...×b, giving aⁿ bⁿ.

(a ÷ b)ⁿ = aⁿ ÷ bⁿ (for b = 0)

Explanation: (a/b)(a/b)...(a/b) (n times) equals (a×a×...×a) ÷ (b×b×...×b) = aⁿ ÷ bⁿ.

Worked Examples

Example 1. Simplify 2³ × 2⁴.
Sol.

Apply the law a^m × aⁿ = a^m+n.

2³ × 2⁴ = 2³⁺⁴

2⁷ = 128

Ans. 128

Example 2. Simplify 5⁶ ÷ 5².
Sol.

Apply the law a^m ÷ aⁿ = a^m−n.

5⁶ ÷ 5² = 5⁶⁻²

5⁴ = 625

Ans. 625

Example 3. Simplify (3²)³.
Sol.

Apply the law (a^m)ⁿ = a^mn.

(3²)³ = 3^2×3

3⁶ = 729

Ans. 729

Example 4. Simplify (2 × 5)³.
Sol.

Apply the law (ab)ⁿ = aⁿ bⁿ.

(2 × 5)³ = 2³ × 5³

2³ × 5³ = 8 × 125

8 × 125 = 1000

Ans. 1000

Example 5. Simplify (6 ÷ 3)².
Sol.

Apply the law (a ÷ b)ⁿ = aⁿ ÷ bⁿ.

(6 ÷ 3)² = 6² ÷ 3²

6² ÷ 3² = 36 ÷ 9

36 ÷ 9 = 4

Ans. 4

Common Points to Remember

• a⁰ = 1 holds only for a = 0.
• a¹ = a always.
• When multiplying powers with the same base, add exponents.
• When dividing powers with the same base, subtract exponents (numerator exponent minus denominator exponent).
• Power of a power multiplies the exponents.
• Power of a product distributes the power to each factor.
• Be careful with signs and zero; check domain restrictions like a = 0 or b = 0 where division is involved.

Practice Problems (for self‐study)

• Simplify 7² × 7³.
• Simplify 4⁵ ÷ 4².
• Simplify (2 × 3)⁴.
• Simplify (10 ÷ 2)³.
• Simplify (5²)³.

Summary: Understand the meaning of base and exponent. Use the five basic laws of exponents to simplify expressions: multiply by adding exponents, divide by subtracting exponents, power of a power multiplies exponents, and powers distribute over multiplication and division. These rules apply to rational (and later real) numbers with attention to non‐zero conditions for divisors and zero exponents.