Quantity A |
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | Quantity B |

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | ||

$a^{-10}$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | $b^{-5}$ |

- Quantity A is greater.
- Quantity B is greater.
- The two quantities are equal.
- The relationship cannot be determined from the information given.

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 7 of the second Quantitative section of Practice Test 1. Those questions testing our knowledge of **Exponents and Roots** can be kind of tricky, but never fear, PrepScholar has got your back!

## Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

We can see **exponents** in both quantities. So this question likely tests our **Exponents and Roots** math skill. Let’s keep what we’ve learned about this skill at the tip of our minds as we approach this question.

## What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

- $a$ is a negative number
- $b$ is a positive number
- We’re comparing $a^{-10}$ to $b^{-5}$

## Develop a Plan

We know that $a$ is **negative** and we have **exponents** in the quantities. **Anytime we see the math concepts of exponents and negative numbers, we should immediately remind ourselves that a negative number raised to an ODD exponent gives a negative result, and that a negative number raised to an EVEN exponent gives a positive result**. We also see even and odd exponents in the quantities, so this tidbit of info is looking like a promising solution method! Let’s use it to analyze the signs of the two quantities.

We see that $a$ is raised to an EVEN power, so therefore Quantity A must be positive. We also know that Quantity B must be positive since $b$ is positive and we know that any positive number raised to an exponent is still a positive number. So with only using the even-odd status of the exponents, we can’t be certain which quantity is greater since all we know is that they’re both positive.

Ah, bummer! That was looking so promising too! Alright, well, the quantities appear to have relatively easy calculations (just raising a number to an exponent), so let’s utilize a **Guess-and-Check** solution method.

## Solve the Question

Let’s choose simple values for $a$ and $b$ so that the calculations aren’t too hard. We know that $a$ must be negative and $b$ must be positive, so let’s try $a=-1$ and $b=1$.

Quantity A | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | Quantity B |

$a^{-10}$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | $b^{-5}$ |

$(-1)^{-10}$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | $1^{-5}$ |

$1$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | $1$ |

Excellent. So we found that in this case the two quantities were equal, which eliminates A and B as answer choices. Good thing we remembered that for Quantity A, $-1$ raised to ANY EVEN NUMBER gives us $1$, and for Quantity B, $1$ raised to ANY number also gives us $1$.

So the right answer is either C or D. Let’s try another pair of numbers for $a$ and $b$. Seeing that we want to get A or B on our next trial to prove that D is correct, let’s try to use a pair of numbers $a$ and $b$ that seem like they’ll give us differing values for our two quantities. After all, we can’t be *too* surprised that $1$ and $-1$ gave us the same values for the two quantities; they’re very similar! So let’s try $a=-1$ and $b=2$.

Quantity A | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | Quantity B |

$a^{-10}$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | $2^{-5}$ |

$(-1)^{-10}$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | $2^{-5}$ |

$1$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | $2^{-5}$ |

Even if we forgot how to evaluate $2^{-5}$, it definitely doesn’t look like it’s $1$, so we could rule out C as an answer choice and choose the only remaining answer, D.

However, we definitely should learn how to evaluate negative exponents, so let’s quickly refresh our memories about how to simplify expressions with negative exponents by using the **Negative Exponent Rule**.

### Concept Refresher – Negative Exponent Rule

We can convert negative exponents into positive exponents by just moving the entire base and exponent from the numerator to the denominator, or vice versa. We can also think of this as taking the **reciprocal** of a number (where we divide $1$ by that number) and then change the exponent to positive. For example:

$$5^{-7} = (1/5)^7$$

OR

$$(1/4)^{-5} = 4^5$$

Now let’s get back to the question at hand.

Applying the Negative Exponent Rule to Quantity B, we get:

$$2^{-5}=1/2^5=1/32$$

$1/32$ is definitely not equal to $1$, so we can eliminate C, and we only have D remaining. **The correct answer is D,
the relationship cannot be determined from the information given**.

## What Did We Learn

Slow and steady wins the race! We ran into a couple of roadblocks, namely going down a fruitless solution path thinking that we could determine the signs of the two quantities to solve this question. However, we never became discouraged, pressing on with a **Guess-and-Check** solution method and eventually getting the right answer. Though it would be nice if we always knew if a certain solution method will work for every type of question, that isn’t the case. So we need to remain flexible and know when to change directions if we’re not making much progress on a solution.

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