So, you were trying to lớn be a good thử nghiệm taker & practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 18 of Section 6 of Practice demo 1. Those questions testing our knowledge of Exponents and Roots can be kind of tricky, but never fear, firmitebg.com 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 khổng lồ think about what type of math knowledge we’ll use lớn 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.
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We see that we want to evaluate $x^-2$, so the presence of an exponent suggests that we’ll use our Exponents và Roots math skill. We also have the sản phẩm of two algebraic expressions is equal lớn $0$, which we recognize from our Solving Quadratic Equations math skill. 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 các mục of the things that we know.
We have the equation: $(x+2)(x-3)=0$We have the inequality: $x>1/2$We want to find the value of $x^-2$Develop a Plan
Well, the question appears khổng lồ be fairly straightforward. Still, let’s make a menu of the plan we’ll use to lớn find our solution.
We have an equation with $x$ in it, so let’s solve it for $x$Make sure that our value for $x$ satisfies our inequalityCalculate the value of $x^-2$Solve the Question
We remember from our Solving Quadratic Equations math skill a special case for when the product of two algebraic expressions is equal to lớn $0$. Specifically, we know that for the sản phẩm of two algebraic expressions lớn be equal lớn $0$, one of the algebraic expressions MUST be equal to $0$.
So from our equation, either $(x+2)=0$, in which case $x=-2$, OR $(x-3)=0$, in which case $x=3$. So from our equation, $x$ can have the values of $-2$ or $3$.
However, we vì chưng have an inequality for $x$, stating that $x$ must be greater than $1/2$. So we can discard the $x=-2$ solution, giving us $x=3$ as the only solution to the equation & inequality.
Excellent! Now all we have to vì chưng is solve $x^-2$, which we now know is $3^-2$. We see that we have a negative exponent, so let’s đánh giá how we simplify negative exponents, if we’ve forgotten how to vày so, with the Negative Exponent Rule.
Concept Refresher – Negative Exponent Rule
We can convert negative exponents into positive exponents by just moving the entire base và exponent from the numerator to lớn 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) & then change the exponent to lớn positive. For example:
$$5^-7 = (1/5)^7$$
OR
$$(1/4)^-5 = 4^5$$
Now let’s get back to lớn the question at hand.
Applying the Negative Exponent Rule lớn simplify $3^-2$, we get:
$$3^-2=1/3^2=1/9$$
The correct answer is D, $1/9$.
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What Did We Learn
Combining completely different math skills is a favorite way for the GRE to increase the difficulty of some of its math questions. However, we were wise lớn realize that we could use our math skills sequentially, by FIRST solving for $x$ và then calculating the value of $x^-2$. Divide & conquer is the way lớn go!
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