List $L$ consists of the number $1, √2, x, \and x^2$, wherein $x>0$, and the selection of the numbers in perform $L$ is $4$.

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Quantity A$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$Quantity B
Quantity A is greater.Quantity B is greater.The two amounts are equal.The connection cannot be identified from the information given

So, you were trying to it is in a an excellent test taker and also practice for the GRE v PowerPrep online. Buuuut then you had actually some questions around the quant section—specifically inquiry 7 of section 4 of practice Test 1. Those questions experimentation our Exponents and Roots knowledge can be type of tricky, but never fear, has obtained your back!

Survey the Question

Let’s search the trouble for clues regarding what it will certainly be testing, as this will help change our minds come think about what type of math expertise we’ll usage to resolve this question. Pay fist to any words that sound math-specific and anything special about what the number look like, and also mark them on your paper.

Since the question provides us the range the a list of numbers, it likely tests ours Numerical methods for describing Data mathematics skill. We also have a square source symbol $(√\;\;\;)$ and an exponent, so we’ll likely utilize what we know about Exponents and also Roots. Let’s keep what we’ve learned about these an abilities at the tip of ours minds together we approach this question.

What perform We Know?

Let’s very closely read with the question and also make a perform of the points that we know.

We have actually a list of four numbers, through two of castle unknown ($x$ and $x^2$)The selection of the list is $4$We want to to compare $x$ come a value

Develop a Plan

Let’s begin with a top-down approach, whereby we will begin with what we’re spring for and work under to the details of what we’re given in this question. We desire to to compare $x$ to the value $2$, for this reason let’s think that what we know around $x$. It’s component of a list of numbers, and we’re given the range of the list. Let’s remind what we know around calculating the selection of a perform of numbers:

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

Since we know the variety is $4$, we deserve to use this info if we likewise figure the end which terms are the minimum and also maximum values. We know two the the four values in this list: $1$ and $√2$. Since $1=√1$ and also $2=√4$, climate we understand that $√2$ have to be somewhere between $1$ and $2$. We don’t understand for sure that $1$ is the minimum worth yet, as perhaps $x$ or $x^2$ can be the minimum value. Let’s look in ~ some values of $x$ and $x^2$ so the we have actually a much better sense that what they could be. Then us can number out i beg your pardon values are the minimum and maximum values and also use the variety to calculate the value of $x$.

Solve the Question

Let’s do a table because that $x$ and $x^2$ values, keeping in mind the $x$ needs to be positive.

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

Well, automatically we notice that $x^2$ is larger than $x$. Hmm, looking in ~ these values for $x$, if we’re walk to gain a range of $4$, climate it looks prefer we’ll need $x$ to be somewhere approximately $2$ or $3$. Otherwise we won’t have a preferably value huge enough to gain a range of $4$. Since $x^2$ values room bigger 보다 the $x$ values, to get a range of $4$, the $x^2$ will certainly be the higher of these 2 values, and likewise be the best value. So that looks prefer $1$ will be the minimum value and $x^2$ will certainly be the best value. Let’s calculate the best value currently using the selection equation:

$\Maximum \Value$$=$$\Range + \Minimum \Value$
$ $$ $$ $
$\Maximum \Value$$=$$4+1$
$ $$ $$ $
$\Maximum \Value$$=$$5$

Since the minimum value is $5$, this provide us $x^2=5$. To gain $x$ we have the right to just take the square root of $5$, offering us $x=√5$. Excellent! Let’s placed that in for quantity A and compare the two amounts now.

When compare an integer come a number under a radical, it’s usually simpler to convert the integer to a radical and also then compare them. Due to the fact that $2=√4$, let’s use that worth for amount B.

So we’re now comparing quantity A $(√5)$ to amount B $(√4)$ The exactly answer is A, amount A is greater.

What Did us Learn

When compare an integer to a number under a radical, us should convert the integer come a radical then continue with the comparison. Also, having a good “number sense” can assist us deal with questions around numerical approaches for relenten data. In this question, it was useful to easily look at a few $x$ values so the we might see that the worth for $x$ needed to be about $2$ or $3$ to get a selection of $4$.

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