If a = -0.3, which of the following is true?

We are given that a = -0.3; what we must keep in mind in this problem is that we are really being tested on what happens when we raise a negative proper fraction to an exponent. We say negative proper fraction because -0.3 = -3/10.

The fact that a = -0.3 is not that important; in fact, we could use any negative proper fraction, such as -1/2 or -1/3 to obtain the correct answer.

Let's look at what happens when a negative proper fraction is raised to an even or odd power.

Rule: When a negative proper fraction is raised to an even-powered or odd-powered exponent, the value increases. If this rule is hard to see, let’s test -1/2. We can start by raising -½ to the even exponent of 2.

(-1/2)^2 =1/4

¼ > -1/2

Clearly, we can see that positive ¼ is greater than -½.

Let’s now check an odd-powered exponent; we can raise -1/2 to an exponent of 3.

(-1/2)^3 = -1/8

-1/8 > -1/2

Although both values are negative, -1/8 is greater than -½.

We can use these rules to answer the question. Since we are raising a negative base to a power, the largest value will be a^2, since that is positive, and a and a^3 are negative. That leaves us with answers B and E. Finally, we know a^3 must be greater than a, thus eliminating answer choice E.

The answer is B.

Hi legendinthewomb,

I think that you mean to say that -0.3 is less than - 0.027 (since -0.3 is further to the 'left' on a Number Line).

GMAT assassins aren't born, they're made,
Rich

yes lol. *facepalms

a^2 is the only positive number, so A,C and D are out. As a is a negative number a^3>a
Answer (B)

it's very tricky! I got into the E trap :D

if we calculate it first: -0,3, -0,27, 0,09
we can figure out that B is the answer

Hi nandetapuri,

It looks like you 'mixed up' the order of the three values. Notice that the largest term in Answer B is "a^2" (which equals +0.09), while the largest term in Answer D is "a" (which equals -0.3). So the answer to your immediate question is NO - Answers B and D are NOT the same.

GMAT assassins aren't born, they're made,
Rich

If a = -0.3, then:
a = -0.3
a² = (-0.3)(-0.3) = 0.09
a³ = (-0.3)(-0.3)(-0.3) = -0.027

If we order these values, we get: -0.3 < -0.027 < 0.09
In other words, a < a³ < a²
Answer:B

Can't believe i got this wrong. I chose E by completely forgetting that -.027 is less than -.3. Hate this aah

\(a = -0.3\) and we need to find the relationship between a, \(a^2\) and \(a^3\)

Let's solve the problem using two methods:

Method 1: Substitution

a = -0.3
\(a^2\) = \((-0.3)^2\) = 0.09
\(a^3\) = \((-0.3)^3\) = -0.027

=> a < \(a^3\) < \(a^2\)

So, Answer will be B

Method 2: Logic

Now \(a^2\) will be positive and will be greater than a and \(a^3\)

As a is a negative fraction between -1 and 0
=> \(a^3\) will become a smaller negative number and will be greater than a
\(a^3\) = -0.027 > -0.3

=> a < \(a^3\) < \(a^2\)

So, Answer will be B
Hope it helps!

Watch the following video to learn the Basics of Exponents

Can someone please explain this ? Read it in the MGMAT Algebra book (Page 39).

-2^4 = Negative

(-2)^4 = Positive

So, how is it that -0.3^even number is even ?

Hi torreadortorment,

(-.3)^(even) would NOT end in an even number. It would be a POSITIVE number though.

GMAT assassins aren't born, they're made,
Rich

Sorry my bad. I meant, how is it that -0.3^even number is positive ? I know why (-2)^4 = Positive. However, isn't -2^4 = Negative>

Hi torreadortorment,

This is actually a Number Property Rule. Almost any number, when raised to an even power, will end in a positive number (the exception is 0).

For example:

2^2 = (2)(2) = 4
(-2)^2 = (-2)(-2) = +4

3^4 = (3)(3)(3)(3) = 81
(-3)^4 = (-3)(-3)(-3)(-3) = 81

When using an EVEN power, the negative terms "cancel out", so the result is always positive.

This holds true even when we use a negative power:

(-2)^(-2) = 1/(-2)(-2) = 1/4

The exception I mentioned earlier:

0^2 = (0)(0) = 0
Etc.

GMAT assassins aren't born, they're made,
Rich

Thanks. I understood that. But the question says -0.3 and not (-0.3).

-0.3 = -1 * 0.3. Hence, whatever happens, it would be -1 * (0.3)^even number, hence the overall result would still be negative. Read this in the MGMAT Algebra book in the exponents chapter.

Hi torreadortorment,

You've misinterpreted the question.

Before substituting in the value for 'A', let's look at 'A' to an even power:

A^2 = (A)(A)
A^4 = (A)(A)(A)(A)
Etc.

NOW, plug in the value for A.... (A = -0.3)....

A^2 = (-0.3)^2 = (-0.3)(-0.3) = +.09
Etc.

IF.... you were given -0.3^2 to start off with, then that calculation WOULD lead you to -.09 (since you have to use PEMDAS rules), but that is NOT what we were given in this question.

GMAT assassins aren't born, they're made,
Rich

Hello could you please tell me why answer D is incorrect.

D) 0.09 > -0.0027 > -0.3
B) -0.3 < -0.027 < 0.09


Aren't both of them same?

As I just got reminded the hard way, I am most accurate with these problems when I use fractions and a number line.

a= -\(\frac{3}{10}\)

a\(^2\) = (-\(\frac{3}{10})^2\) = \(\frac{9}{100}\), or, when the numbers are unwieldy, "some +"

a\(^3\) = (-\(\frac{3}{10}\))\(^3\) = -\(\frac{27}{1000}\)

___-\(\frac{3}{10}\) ___-\(\frac{27}{1000}\)__0___\(\frac{9}{100}\)___1_____

____\(a\)______\(a^3\)____|____\(a^2\)____|_____

\(a < a^3 < a^2\)

Answer B