If $ A = 8^3 imes 5^4 $ and $ B = 8^5 imes 5^3 $, then the value of $ A imes B $ is:

Question

If $ A = 8^3 	imes 5^4 $ and $ B = 8^5 	imes 5^3 $, then the value of $ A 	imes B $ is:

Examoogle · Accepted Answer

The correct answer is Option A: $2^{24} 	imes 5^7$. Use the property of exponents to add powers of the same base. Here, $A 	imes B = (8^3 	imes 5^4) 	imes (8^5 	imes 5^3) = (8^3 	imes 8^5) 	imes (5^4 	imes 5^3) = 8^{(3+5)} 	imes 5^{(4+3)} = 8^8 	imes 5^7$. Since $8 = 2^3$, replace 8 with $2^3$ to get $(2^3)^8 = 2^{24}$, resulting in $2^{24} 	imes 5^7$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Use the property of exponents to add powers of the same base. Here, $A 	imes B = (8^3 	imes 5^4) 	imes (8^5 	imes 5^3) = (8^3 	imes 8^5) 	imes (5^4 	imes 5^3) = 8^{(3+5)} 	imes 5^{(4+3)} = 8^8 	imes 5^7$. Since $8 = 2^3$, replace 8 with $2^3$ to get $(2^3)^8 = 2^{24}$, resulting in $2^{24} 	imes 5^7$.

Formula Used: $$a^m 	imes a^n = a^{m+n}$$

Given: A = 8^3 × 5^4, B = 8^5 × 5^3

Steps: Setup the multiplication: Substitute the expressions for $A$ and $B$ into the product $A 	imes B$.. Group bases using index laws: Use the associative property to group the same bases together: $8^3 	imes 8^5$ and $5^4 	imes 5^3$.. Express 8 as a power of 2: Since $8 = 2^3$, substitute this into the expression and apply the power-of-a-power rule $(x^m)^n = x^{m 	imes n}$.

Answer: A is correct because the product evaluates to $2^{24} 	imes 5^7$ using the laws of exponents.

Exam Tip: This type of simplification using index laws is standard in competitive exams like RRB, SSC, and Banking to test conceptual clarity on powers.

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