List $L$ consists of the numbers $1, √2, x, \and x^2$, where $x>0$, and the range of the numbers in list $L$ is $4$.

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

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

$x$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | $2$ |

- 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 Section 4 of Practice Test 1. Those questions testing our **Exponents and Roots** knowledge 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 your paper.

Since the question gives us the **range** of a list of numbers, it likely tests our **Numerical Methods for Describing Data** math skill. We also have a **square root symbol $(√{\;\;\;})$** and an **exponent**, so we’ll likely utilize what we know about **Exponents and Roots**. Let’s keep what we’ve learned about these skills 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.

- We have a list of four numbers, with two of them unknown ($x$ and $x^2$)
- The range of the list is $4$
- We want to compare $x$ to a value

## Develop a Plan

Let’s start with a top-down approach, where we will begin with what we’re looking for and work down to the details of what we’re given in this question. We want to compare $x$ to the value $2$, so let’s think of what we know about $x$. It’s part of a list of numbers, and we’re given the **range** of that list. Let’s recall what we know about calculating the range of a list of numbers:

$$\Range = \Maximum \Value – \Minimum \Value$$

Since we know the range is $4$, we can use this info if we also figure out which terms are the minimum and maximum values. We know two of the four values in this list: $1$ and $√2$. Since $1=√1$ and $2=√4$, then we know that $√2$ must be somewhere between $1$ and $2$. We don’t know for sure that $1$ is the minimum value yet, as perhaps $x$ or $x^2$ could be the minimum value. Let’s look at some values of $x$ and $x^2$ so that we have a better sense of what they could be. Then we can figure out which values are the minimum and maximum values and use the range to calculate the value of $x$.

## Solve the Question

Let’s make a table for $x$ and $x^2$ values, keeping in mind that $x$ has to be positive.

$$\table x,x^2; 0, 0; 1, 1; 2, 4; 3, 9; 4, 16; 5, 25$$

Well, immediately we notice that $x^2$ is larger than $x$. Hmm, looking at these values for $x$, if we’re going to get a range of $4$, then it looks like we’ll need $x$ to be somewhere around $2$ or $3$. Otherwise we won’t have a maximum value large enough to get a range of $4$. Since $x^2$ values are bigger than the $x$ values, to get a range of $4$, the $x^2$ will be the greater of these two values, and also be the maximum value. So it looks like **$1$ will be the minimum value and $x^2$ will be the maximum value**. Let’s calculate the maximum value now using the range equation:

$\Maximum \Value$ | $=$ | $\Range + \Minimum \Value$ |

$ $ | $ $ | $ $ |

$\Maximum \Value$ | $=$ | $4+1$ |

$ $ | $ $ | $ $ |

$\Maximum \Value$ | $=$ | $5$ |

Since the minimum value is $5$, this give us $x^2=5$. To get $x$ we can just take the square root of $5$, giving us $x=√5$. Excellent! Let’s put that in for Quantity A and compare the two quantities now.

**When comparing an integer to a number under a radical, it’s usually easier to convert the integer to a radical and then compare them**. Since $2=√4$, let’s use that value for Quantity B.

So we’re now comparing Quantity A $(√5)$ to Quantity B $(√4)$ **The correct answer is A, Quantity A is greater**.

## What Did We Learn

When comparing an integer to a number under a radical, we should convert the integer to a radical then continue with the comparison. Also, having a good “number sense” can help us solve questions about numerical methods for describing data. In this question, it was helpful to quickly look at a few $x$ values so that we could see that the value for $x$ needed to be around $2$ or $3$ to get a range of $4$.

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