Question

Assertion: If 10^a × 100^2a = 1000⁵ then a = 3

Reason: (a^m)ⁿ = a^mn

Options

• Both A and R are true and R is the correct reason for A.

• Both A and R are true but R is the incorrect reason for A.

• A is true but R is false.

• A is false but R is true.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

Let’s analyze this step by step:

Assertion:

If 10^a × 100^2a = 1000⁵, then a = 3

Let’s simplify both sides using exponent laws:

Step 1: Convert all terms to powers of 10

• 100 = 10² → 100^2a = (10²)^2a = 10^4a
• 1000 = 10³ → 1000⁵ = (10³)⁵ = 10¹⁵

Now rewrite the equation:

10^a × 10^4a = 10¹⁵

⇒ 10^a+4a = 10¹⁵

⇒ 10^5a = 10¹⁵

⇒ a = `15/5` = 3

Assertion is true

Reason:

(a^m)ⁿ = a^mn

This is a valid law of exponents.

Reason is true

And the Reason (i.e., (a^m)ⁿ = a^mn) was used to simplify 100^2a = (10²)^2a = 10^4a, so it directly supports the Assertion.