What Are Exponents

Exponent's questions are a great way to improve your understanding of powers and exponents. These questions are based on the latest CBSE and ICSE Class 8 syllabus and are useful for school tests and exams. With regular practice, you can become confident in solving exponent-related problems.

Let’s start by understanding the idea:

Suppose you have a number 'a'.

• If you add it again and again:
  • a + a = 2a
  • a + a + a = 3a
  • Repeating this n times gives: na

Now, let’s talk about multiplying the same number repeatedly:

• a × a = a² (a squared)
• a × a × a = a³ (a cubed)
• If you multiply it n times, it becomes: aⁿ

This process is called exponentiation.

In aⁿ,

• a is the base (the number being multiplied)
• n is the exponent or power (how many times the base is multiplied)

We read aⁿ as "a raised to the power n".

Laws of Exponents

1. Multiplying powers with the same base:
   If the base is the same, just add the powers.
   Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
   Example: 2³ × 2² = 2⁵ = 32
2. Dividing powers with the same base:
   If the base is the same, subtract the powers.
   Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
   Example: 5⁴ ÷ 5² = 5² = 25
3. Multiplying powers with the same power but different bases:
   Multiply the bases, then apply the power.
   Rule: aᵐ × bᵐ = (a × b)ᵐ
   Example: 2² × 3² = (2 × 3)² = 6² = 36
4. Dividing powers with the same power but different bases:
   Divide the bases, then apply the power.
   Rule: aᵐ ÷ bᵐ = (a ÷ b)ᵐ
   Example: 6² ÷ 3² = (6 ÷ 3)² = 2² = 4
5. Negative exponents:
   A negative power means the reciprocal (1 over the number with a positive power).
   Rule: a⁻ᵐ = 1 / aᵐ
   Example: 2⁻³ = 1 / 2³ = 1/8
6. Zero exponent:
   Any number (except 0) raised to the power 0 is always 1.
   Rule: a⁰ = 1
   Example: 7⁰ = 1
7. Power of a power:
   When a power is raised to another power, multiply the powers.
   Rule: (aᵐ)ⁿ = aᵐ×ⁿ
   Example: (3²)³ = 3⁶ = 729
8. Fractional exponents (roots):
   A power like aᵐ/ⁿ means take the n-th root of aᵐ.
   Rule: aᵐ/ⁿ = ⁿ√(aᵐ)
   Example: 8¹/³ = ∛8 = 2

To learn more about exponents and powers, click on the link.

1. Simplify: 2³ × 2⁴
   Solution:
   = 2^(3 + 4)
   = 2⁷
   = 128
2. Simplify: (5²)³
   Solution:
   = 5^(2 × 3)
   = 5⁶
   = 15,625
3. Simplify: (3⁴ × 3²) ÷ 3³
   Solution:
   = 3^(4 + 2 - 3)
   = 3³
   = 27
4. Find the value of: 10⁰
   Solution:
   Any non-zero number raised to the power 0 is 1.
   So, 10⁰ = 1
5. Simplify: (2⁴) ÷ (2²)
   Solution:
   = 2^(4 - 2)
   = 2²
   = 4
6. Simplify: (4²) × (2³)
   Solution:
   = 16 × 8
   = 128
7. Simplify: (3²)²
   Solution:
   = 3^(2 × 2)
   = 3⁴
   = 81
8. Express 1/8 as a power of 2
   Solution:
   8 = 2³, so
   1/8 = 1/(2³) = 2⁻³
   Answer: 2⁻³
9. A number divides (–10)⁻¹ and gives (–2)⁻¹ as the result. What is the number?

Solution:

Let the number be x.

We are given:
(–10)⁻¹ ÷ x = (–2)⁻¹

Now solve step by step:

⇒ –1/10 ÷ x = –1/2
⇒ –1/10 × 1/x = –1/2
⇒ –1/(10x) = –1/2
Now cancel the minus signs on both sides:

⇒ 1/(10x) = 1/2
⇒ 10x = 2
⇒ x = 2/10
⇒ x = 1/5

Final Answer:
x = 1/5 or 5⁻¹

10. A number divides (–8)⁻¹ and gives (–4)⁻¹ as the result. What is the number?

Solution:

Let the number be x.

We are given:
(–8)⁻¹ ÷ x = (–4)⁻¹

Now solve it step by step:

⇒ –1/8 ÷ x = –1/4
⇒ –1/8 × 1/x = –1/4
⇒ –1/(8x) = –1/4
(Cancel the minus signs on both sides)

⇒ 1/(8x) = 1/4
⇒ 8x = 4
⇒ x = 4/8
⇒ x = 1/2

Final Answer:
x = 1/2 or 2⁻¹

Related Articles

Continue to visit our website for more Maths lessons presented in a clear and understandable way. Also, register to get comprehensive assistance for Maths preparation with video lessons, notes, tips, and other study materials.