munaf wrote:10) 2^x - 2^x-2 = 3(2^13), what is x?
a. 9
b. 11
c. 13
d. 15
e. 17
PLEASE EXPLAIN HOW TO SOLVE THIS
OA-POST SOME DISCUSSION
The key to solving this type of question is the basic exponent rule:
x^a * x^b = x^(a+b)
and factoring out powers of x to get the expression in that form.
Let's look an an example using numbers instead of variables:
2^7 + 2^9 = ?
It's impossible to add terms unless they have the same base AND the same power. Accordingly, we need to express both terms similarly.
Using the above-noted rule, we know that:
2^9 = 2^7 * 2^2
So we can rewrite the entire expression as:
2^7 + 2^7 * 2^2 = 1(2^7) + 4(2^7) = 5(2^7)
Basically, we rewrite all the terms with the same power as the term with the smallest exponent.
Some more examples:
3^5 - 3^4 = 3(3^4) - 1(3^4) = 2(3^4)
7^8 + 7^6 = (7^2)(7^6) + 7^6 = 49(7^6) + 1(7^6) = 50(7^6)
Applying that reasoning to this question:
2^x - 2^x-2 = 3(2^13)
we see that our final exponent is 13; therefore, we want to set our lowest exponent to 13.
x-2 is certainly smaller than x, so if we let x-2=13 we get:
2^15 - 2^13 = (2^2)(2^13) - 2^13 = 4(2^13) - 2^13 = 3(2^13).. bingo!
Now, if we were really confident we wouldn't have actually done all that math and would have just solved for:
x-2=13
and chosen x=15 as the correct answer.
