jaber wrote:
Quantity A |
Quantity B |
\((0.82^2)*(0.82^3)\) |
\((0.82^6)\) |
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
I was thinking that the correct answer is B because the exponent of (0.82^6) is 6 which mean is multiplied by itself 6 times which is bigger than (0.82^5) which has the same base and only multiplied by itself 5 times only , that what I thought but I found that the correct answer is A !!!
is it because decimals and fractions power smallest number would be bigger ? unlike the integer number ??
I mean when we compare (2^5) with (2^6) we clearly see that
(2^5) = 32
(2^6) = 64
which means without doubt that (2^6) with the bigger exponents is bigger since the bases are the same in the two values , why the case in the question is totally different ???
Please help
Thank you very much
Remind the following properties.
Case 1: \(0<x<1\)
We have \(\cdots > 1/x^2 > 1/x > 1 > \sqrt{x} > x > x^2 > x^3 > \cdots\).
When a base is between \(0\) and \(1\), as exponents grow, they get smaller.
Case 2: \(x>1\)
We have \(\cdots < 1/x^2 < 1/x < 1 < \sqrt{x} < x < x^2 < x^3 < \cdots\).
When a base is greater than \(1\), as exponents grow, they get bigger.
\((a^x)^y = a^{xy}\) and \(a^x \cdot a^y = a^{x+y}\).
Since we have \(5 < 6\) and \(0 < 0.82 < 1\), we have \((0.82)^5 > (0.82)^6\).
Thus, we have \(A = (0.82)^2 \cdot (0.82)^3 = (0.82)^5 > B = (0.82)^6\).
Therefore, A is the right answer.