Find the value of a. Given a=\(\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}....\) GMAT Problem Solving

Question: If a=\(\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}....\), What is the value of a?

1. 0
2. \(\sqrt{3}\)
3. 3
4. 2.9
5. Cannot be determined

Correct Answer: C

Solution and Explanation
Approach Solution 1:
As per the question, a=\(\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}....\),

This implies that a=\(\sqrt{3(\sqrt{3\sqrt{3\sqrt{3...)}}}}\),

Now, as the expression under the square root extends infinitely, then the expression in brackets would equal to itself. So we can safely replace it with a and rewrite the given expression as a=\(\sqrt{3a}\)

Since there is a square root on the right side of the equation, we need to square the equation on both sides to remove the root.

Squaring the equation: \(a^2=3a\)

This implies a=0 or a=3

Since, a equals zero does not satisfy the equation, then we have to consider the value of a=3.

Hence, the required value of the above problem is 3.

Therefore option C is the correct answer.

Approach Solution 2:
In this solution, What is found here is that, if you have a calculator in mind, then we tend to avoid the algebraic approach. We have followed the algebraic approach.

As per the given question, \(a=\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}\)

Considering the square root, we get,

\(a=\sqrt{3\sqrt{3\sqrt{3^1*3\frac{1}{2}}}}\)

\(a=\sqrt{3\sqrt{3\sqrt{3\frac{3}{2}}}}\)

\(a=\sqrt{3\sqrt{3*3\frac{3}{2}*\frac{1}{2}}}\)

\(a=\sqrt{3\sqrt{3^1*3^\frac{3}{4}}}\)

Removing the square root

\(a=\sqrt{3\sqrt{3\frac{7}{4}}}\)

\(a=\sqrt{3*3^7/_4*^1/ _2}\)

\(a=\sqrt{3^\frac{15}{8}}\)

\(a=3^\frac{15}{8}*^\frac{1}{2}\)

\(a=3^ \frac{15}{16}\)

For n (Infinite) number of occurrences, the equation would be built up as follows. Considering n=15, n+1=16.

\(a=3^\frac{n}{n}+^1\)

We may deem here n≈n+1

\(a=3^1\)
a=3

Hence, the correct option is C.

Approach Solution 3:

a= √3√3√3√3.. --> a= √3(√3√3√3...). Now, as the expression under the square root extends infinitely, then expression in brackets would equal to a itself, so we can safely replace it with a and rewrite the given expression as a= √3a.

Square it: a^2= 3a-->a= 0 or a=3--> since a= 0 does not satisfy given expression, then we have only one solution: a= 3.

“If a=\(\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}....\), What is the value of a?”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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