Which of the Following is Equal to x^18 For All Positive Values of x? GMAT Problem Solving

Question: Which of the following is equal to x^18 for all positive values of x?

1. x^9 + x^9
2. (x^2)^9
3. (x^9)^9
4. (x^3)^15
5. x^4 / x^22

Correct Answer: B

Solution and Explanation:
Approach Solution 1:

The problem statement asks to find what is equal to x^18 and it is given that the value of x is always positive.
We have to check each one of the options to see if they are equal to the value of x^18.
Option 1,

\(x^9\)+\(x^9\)
Here x9 is summed up two times therefore the value will become,
2*\(x^9\) = 2\(x^9\)
This is not equal to x^18.
Coming to option 2,
\((x^2)^9\) ,
There is a property of exponents that \((x^n)^m\) = \((x^m)^n\) = \(x^{m*n}\)
\((x^2)^9\)= \(x^{2*9}\) = \(x^{18}\)
Therefore this is equal to x^18.
Coming to option 3,
\((x^9)^9\)
Using the same property of exponents we get,
\((x^9)^9\) = \(x^{9*9}\) = \(x^{81}\)
This is not equal to x^18
Coming to option 4,
\((x^3)^{15}\)
Using the same property of exponents,
(x3)15 = \(x^{3*15}\) = \(x^{45}\)
This is not equal to x^18
Coming to option 5,
\(\frac{x^4}{x^{22}}\)
There is a property of exponent division that says,
\(\frac{a^n}{a^{m}}\) = \(a^{n-m}\)
Using it we get,
\(\frac{x^4}{x^{22}}\) = \(a^{4-22}\) = \(x^{-18}\)
This is also not equal to x^18.

Approach Solution 2:

The problem statement states that
Given:

• the value of x is always positive

Find out:

• x^18 equals to what.

We can solve the question in a quicker and easier way as follows:

We can write x^18 as (x^2)^9. 

Therefore, x^18 equals to (x^2)^9.

Approach Solution 3:

The problem statement states that
Given:

• the value of x is always positive

Find out:

• x^18 equals to what.

Let’s use the exponents rules and solve the sum with the help of a simpler example. Instead of x^18, just derive the value of 2^6.

Let, x=2

2^4=64

Let's analyse the answer choices that there are some 9's. That is it is half of the 18 from the question stem. We need to replace the 18 with a 6 so that each time there's a 9 in an answer choice, then replace it with half of 6, so 3.

A. 2^3+2^3 = 8+8=16 ... Irrelevant.

B. (2^2)3=4^3=64 ... Correct.

C. (2^3)^3=6^3 ... 6^3 can not be equal to 4^3 and 4^3 worked, so 6^3 does not support the question. Irrelevant.

D. (2^3)^15 ...  Eliminated.

E. 2^4/2^22... it is less than 1, thereby eliminated.

 “Which of the following is equal to x^18 for all positive values of x?”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This GMAT problem solving question has been taken from The Official Guide for GMAT Quantitative Review. GMAT Problem Solving questions assess the efficiency level of candidates in solving quantitative problems. GMAT Quant practice papers consist of various types of quantitative problems that help to enhance the mathematical understanding of the candidates.

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