If a² = 12, what is the value of a⁴?

Introduction / Context:
This question tests your understanding of powers of a variable and how to manipulate exponents using basic algebraic rules. Such problems are very common in aptitude tests and algebra sections where quick simplification is required.

Given Data / Assumptions:

• We are given a² = 12.
• We must find the value of a⁴.
• No additional constraints on a are given, so we work algebraically with exponents.

Concept / Approach:
We use the law of indices that says a⁴ = (a²)². Instead of finding a explicitly, we can square the given expression. This avoids square roots and keeps the calculation simple and exact. The key idea is to recognize that if we already know a², then a⁴ is simply the square of that known quantity.

Step-by-Step Solution:
Step 1: Start from the given relation a² = 12.Step 2: To obtain a⁴, square both sides of the equation: (a²)² = 12².Step 3: Simplify the left hand side: (a²)² = a⁴.Step 4: Calculate the right hand side: 12² = 12 * 12 = 144.Step 5: Therefore, a⁴ = 144.

Verification / Alternative check:
If you want to verify, you can write a = √12. Then a⁴ = (√12)⁴ = (12)² / (since √12 squared is 12) which again gives 144, confirming the previous result.Both direct exponent rules and substitution using square roots give the same answer.

Why Other Options Are Wrong:
72 would correspond to 12 * 6 and does not match 12².36 is equal to 6², but we are not told that a² = 6, so 36 is not correct here.24 is 12 * 2 and again does not follow from squaring 12.

Common Pitfalls:
Some students mistakenly think a⁴ = a² * 4 or multiply the exponent by the base, which is incorrect.Another common error is to miscalculate 12² as 124 or 142 due to haste.Always remember that (a²)² means a raised to the power 2 * 2, which is 4, and the numerical side must be squared properly.

Final Answer:
The value of a⁴ is 144.