Fraction exponents?

Question

If the equation y=2^x is graphed, which of the following values of x would produce a point closest to the x-axis?

a. 1/4
b. 3/4
c. 5/3
d. 8/3

Is this more a logic problem than solving an equation?

Answer 1

If you are allowed negative numbers, I'd pick one of those. Given your choices, the best answer is:

a. 1/4

The graph goes up exponentially and is "non-decreasing".
x = 0 returns y = 1
x = 1 returns y = 2
x = 2 returns y = 4
x = 3 returns y = 8

The question is really asking you to figure out what they mean by "closest to the x-axis". That would be the value with the smallest absolute value for y... since it is an increasing function, a smaller x corresponds to a smaller y.

So the closer to 0 the better, at least for positive numbers.

The smallest number is:
a. 1/4

Answer 2

a) 1/4
The reason for this is because fraction exponents are equivalent to surds. (The reason for this, in turn, is to make them consistent with the laws of exponents. See http://www.intmath.com/Exponents-radicals/2_Fractional-exponent-laws.php
So 2^(1/4) is the fourth root of 2, approx 1.189
2^(3/4) is the above number, cubed, approx 1.682...
2^(5/3) is the cube root of 2, raised to the fifth power, approx 3.175...
2^(8/8) is the cube root of 3, raised to the eighth power, approx 6.3496...
In general, A^(1/n) is the nth root of A, and A^(m/n) is the nth root of A, raised to the mth power.
By the way, if you graph y = N^x, with N any positive constant, you'll see that it is a continuous curve. This means that we can have not only fractional exponents, but irrational numbers as exponents as well. This is in fact the main idea behind logs.
You ask if this is more a logic problem than solving an equation. In answering it, I did not use formal logic (a branch of discrete math) but I did use my sense of numeracy, that is, of how numbers work, and it appears that most other answerers did as well.

Answer 3

1/4

Answer 4

A. 1/4 because the fourth root of 2 is smaller than any of the other numbers.

Answer 5

Find the common demoninator
a. 1/4= 3/12
b. 3/4= 9/12
c. 5/3= 20/12
d. 8/3= 24/12
Soo.... a is the closest and d is the furtherest

Answer 6

Simplify from the interior out. keep in mind that a / (b / c) = ca / b and (a^-n) = (a million / a^n). a million / ((27a^4/3) / (b^-a million/2c^5/6)) = (b^-a million/2c^5/6) / (27a^4/3) = (c^5/6) / ((b^a million/2)(27a^4/3)) The fractional exponents characterize taking the b-th root of the flexibility a as in: x^(a/b)

Answer 7

its 1/4
4th root of 2 to the power of 1, its closest to 0

Answer 8

the smallest x value, 1)1/4