algebra strategy guide p. 125 #3

[gmatismyfriend]

When you rephrase you get (3/4)^(3/4) is greater than (3/4). I picked false because when you compare exponents, .75 is less than 1.

I thought when the bases are the same you compare exponents. Please help

[jnelson0612]

When you rephrase you get (3/4)^(3/4) is greater than (3/4). I picked false because when you compare exponents, .75 is less than 1.

I thought when the bases are the same you compare exponents. Please help

Let's carefully break this down:
1) Your exponent is 3/4. What does that mean? Let's think about it using an easier fraction exponent, 1/2. The exponent 1/2 means that I am taking the square root of a number. The exponent 1/3 means that I am taking the cube root of a number. The exponent 3/4 means that I am cubing the number and then taking the fourth root of the number.

2) For a normal positive integer, when I take it to a fractional exponent I make the number smaller. For instance, 4^(1/2) becomes 2, the square root of 4.

3) BUT, fraction bases behave differently. For instance, if I take the square root of 1/4, or (1/4)^1/2, the number becomes 1/2. When I take a fraction to a fraction exponent (as long as that fraction and the fraction exponent are both between 0 and 1) I make the number LARGER.

4) Because of this, (3/4)^3/4 is actually LARGER than 3/4. This problem is testing your knowledge of this principle.

[gmatismyfriend]

Ok I see...but can you just simplify the terms to make them the same, and then compare the exponents of each term?

i.e 0.75 < 1

[RonPurewal]

Ok I see...but can you just simplify the terms to make them the same, and then compare the exponents of each term?

i.e 0.75 < 1

No, because the behavior is the opposite when the base number (the number that's raised to the powers) is between 0 and 1.

You don't strictly have to think about the fractional exponents, though -- the ordering is the same as it is for non-fractional exponents.
E.g., 2^(1/2) < 2^(3/4) < 2 < 2^2 < 2^3, etc.
but
(1/2)^3 < (1/2)^2 < 1/2 < (1/2)^(3/4) < (1/2)^(1/2), etc.

In other words, you definitely have to think about whether the base number is a "fraction" or not. But, the comparison between x^1 and x^(3/4) will go the same way as the comparison between x^2 and x^1.