If √24 +√12 = 2a√3, what is the value of a?

Rationale

2√3 is equivalent to √2 + 1 when simplified.

To solve the equation √24 + √12 = 2a√3, we first simplify the left side. √24 can be rewritten as √(4 × 6) = 2√6, and √12 can be rewritten as √(4 × 3) = 2√3. Thus, √24 + √12 = 2√6 + 2√3, which simplifies to 2(√6 + √3). Setting this equal to 2a√3 allows us to find that a = √6/√3 = √2, and thus a = √2 + 1.

A (3) YOUR ANSWER

This choice does not satisfy the original equation. Plugging a = 3 into 2a√3 gives 6√3, which is not equal to the simplified left side of the equation (2√6 + 2√3).

B (6) YOUR ANSWER

Selecting a = 6 leads to 12√3, which greatly exceeds the sum of the square roots on the left side. Hence, it cannot be the correct value for a.

C (√3) YOUR ANSWER

If a = √3, then 2a√3 becomes 2√3 * √3 = 6, which does not match the left side's value of 2√6 + 2√3 upon simplification.

D (2√3) YOUR ANSWER

Choosing a = 2√3 results in 4√3, which again does not equal the left side. The simplified expression does not support this value either.

E (√2+1) CORRECT

This choice equates to 2a√3 as it simplifies correctly when substituted back into the equation, confirming that it is indeed the solution.

Conclusion

The problem requires simplifying the left-hand side and equating it to the right-hand side. After evaluating all choices, only a = √2 + 1 satisfies the equation, demonstrating the importance of simplification and accurate substitution in algebraic problems.

If √24 +√12 = 2a√3, what is the value of a?

2√3 is equivalent to √2 + 1 when simplified.

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