** Exponents**

Every Data Sufficiency (DS) question can have 1 of 5 possible answers, and the possible answers are the same for every DS question. All DS questions are really two questions in one. You’re given a general problem, followed by two supplemental statements. The real problem is to evaluate each of the two statements independently to determine their individual sufficiency to solve the original general problem. As you determine the sufficiency of these statements, you are able to decide which of the following 5 answers to the Data Sufficiency question is the correct answer:

**A. Statement 1 is sufficient, but statement 2 is not.**

**B. Statement 1 is not sufficient, but statement 2 is.**

**C. Neither statement 1 nor statement 2 is sufficient, but together they are.**

**D. Both statement 1 and statement 2 are individually sufficient.**

**E. Neither statement 1 nor statement 2 is sufficient, and together they are not sufficient.**

The diagram above gives a pictorial representation of the 5 Data Sufficiency answers. The two statement numbers are shown in blue at the top of the first two columns. Note that green means sufficient, and red means not sufficient. As an example, if both statements 1 and 2 are individually sufficient, then the correct answer is D. For more information on the 5 DS answers, see our videos.

We illustrate these answers for exponent problems with the following **3 examples**. The first example provides 5 different versions of supplementary statements for the same general problem, and each of these 5 versions yields a different one of the 5 possible answers. The answers to these 5 versions are explained immediately after the problem. After the first example, we then give 2 additional examples with just one version of the supplementary statements.

[Notice that the 5 versions of the supplementary statements provide examples of the 5 different answers, A – E. However, they are not given in A – E order.]

**Example 1)**

\(\begin{align}x^{100} =\end{align}\) ?

Version I.

(1) \(\begin{align}x^5 = 243\end{align}\)

(2) \(\begin{align}x^{-3} = 1/27\end{align}\)

Version II.

(1) \(\begin{align}x^{10} = y^2\end{align}\)

(2) \(\begin{align}x^5 = 243\end{align}\)

Version III.

(1) \(\begin{align}x^{10} = y^2\end{align}\)

(2) \(\begin{align}x^{100} = (x^{10})^{10}\end{align}\)

Version IV.

(1) \(\begin{align}x^{-3} = 1/27\end{align}\)

(2) \(\begin{align}x^{100} = (x^{10})^{10}\end{align}\)

Version V.

(1) \(\begin{align}x^{10} = y^2\end{align}\)

(2) \(\begin{align}y = 243\end{align}\)

**Answers: [ Version I: D Version II: B Version III: E Version IV: A Version V: C ]**

**Explanation: **

For practical purposes \(\begin{align}x^{100}\end{align}\)is much too large to calculate, regardless of the value of *x* (unless it’s 0, 1, -1, or a decimal fraction like .0000001). However, the problem is not to actually calculate \(\begin{align}x^{100},\end{align}\) but to discover whether the data given is sufficient to calculate it. You can do that by finding whether the statements give us the value of any power of *x*. If so, then we have sufficient data to calculate \(\begin{align}x^{100}\end{align}\)because if we have the value of some power of *x*, then we can calculate *x*, and from this we (or a computer) can calculate \(\begin{align}x^{100}.\end{align}\)

**Version I: Answer D – Both statement 1 and statement 2 are individually sufficient.**

(1) \(\begin{align}x^5 = 243\end{align}\)

**Full information. If we have the value of any non-zero power of x we can calculate x, and so we can calculate \(\begin{align}x^{100}.\end{align}\) Sufficient.**

(2) \(\begin{align}x^{-3} = 1/27\end{align}\)

**Full information. If we have the value of any non-zero power of x we can calculate x, and so we can calculate \(\begin{align}x^{100}.\end{align}\) Sufficient.**

**Version II: Answer B – Statement 1 is not sufficient, but statement 2 is.**

(1) \(\begin{align}x^{10} = y^2\end{align}\)

**Partial information. This gives us one equation in two unknowns. We need at least one more equation in these unknowns. Not sufficient.**

(2) \(\begin{align}x^5 = 243\end{align}\)

**Full information. If we have the value of any non-zero power of x we can calculate x, and so we can calculate \(\begin{align}x^{100}.\end{align}\) Sufficient.**

**Version III: Answer E – Neither statement 1 nor statement 2 is sufficient, and together they are not sufficient.**

(1) \(\begin{align}x^{10} = y^2\end{align}\)

**Partial information. This gives us one equation in two unknowns. We need at least one more equation in these unknowns. Not sufficient.**

(2) \(\begin{align}x^{100} = (x^{10})^{10}\end{align}\)

**No information. This is a simple manipulation of exponents and does not give us a specific value for a power of x. Not sufficient.**

**Alone, neither of these statements is sufficient, and together they are not sufficient, because the first gives us a relationship between x and a new variable, y, but the second gives us no specific information about x or y.**

**Version IV: Answer A – Statement 1 is sufficient, but statement 2 is not.**

(1) \(\begin{align}x^{-3} = 1/27\end{align}\)

**Full information. If we have the value of any power of x we can calculate x, and so we can calculate \(\begin{align}x^{100}.\end{align}\) Sufficient.**

(2) \(\begin{align}x^{100} = (x^{10})^{10}\end{align}\)

**No information. This is a simple manipulation of exponents and does not give us a specific value for a power of x. Not sufficient.**

**Version V: Answer C – Neither statement 1 nor statement 2 is sufficient, but together they are.**

(1) \(\begin{align}x^{10} = y^2\end{align}\)

**Partial information. This gives us one equation in two unknowns. We need at least one more equation in these unknowns. Not sufficient.**

(2) \(\begin{align}y = 243\end{align}\)

**Partial information. This gives us an equation in y, but we need a connection of y to x. Not sufficient.**

**Alone, neither of these statements is sufficient, but together they are sufficient, because the two statements give us two equations in two unknowns, and from this we can calculate the value of x.**

**To practice more problems like this, click** here.

**Example 2)**

If *a* and *b* are integers, and \(\begin{align}9^{a+b} = 27^{a-b}\end{align}\), what are the values of *a* and* b*?

(1) Doubling *a* and adding *b* yields 11.

(2) *a* + *b* = 6

Note that \(\begin{align}3^2 = 9\end{align}\), and \(\begin{align}3^3 = 27\end{align}\), so the equation can be rewritten as \(\begin{align}(3^2)^{a+b} = (3^3)^{a-b}\end{align}\), which becomes \(\begin{align}3^{2a+2b} = 3^{3a-3b}\end{align}\), and therefore \(\begin{align}2a + 2b = 3a – 3b\end{align}\), or \(\begin{align}a = 5b\end{align}\). Therefore, we already have one equation in the two unknowns, *a* and* b*. We just need one more equation.

**Answer: D – Both statement 1 and statement 2 are individually sufficient.**

**Explanation:**

(1) Doubling a and adding b yields 11.

**Full information. This is the equation 2a + b = 11. Together with a = 5b, we have two equations in two unknowns, and solving we find that a = 5 and b = 1. Sufficient.**

(2) a + b = 6.

**Full information. Together with a = 5b, we get a = 5 and b = 1. Sufficient.**

**Example 3)**

What is x to the 4th power?

(1) \(\begin{align}x^3 = y^6\end{align}\)

(2) \(\begin{align}y^2 = 8\end{align}\)

**Answer: C – Neither statement 1 nor statement 2 is sufficient, but together they are.**

**Explanation:**

(1) \(\begin{align}x^3 = y^6\end{align}\)

**Partial information. This gives us a single equation in 2 unknowns. We need ****another equation in x and y. Not sufficient.**

(2) \(\begin{align}y^2 = 8\end{align}\)

**Partial information. This gives us a single equation in a new unknown. We need ****another equation in x and y. Not sufficient.**

**Alone, neither of these statements is sufficient, but together they are sufficient, because we now ****have 2 equations in 2 unknowns. If \(\begin{align}y^2 = 8\end{align}\), then \(\begin{align}y^6 = (y^2)^{3} = 8^3\end{align}\), so \(\begin{align}x^3 = 8^3\end{align}\), and \(\begin{align}x = 8\end{align}\).**

**To practice more of these types of problems, click** here.