are these two statements true?
(a/b)^x = a^x - b^x
and
(x^a + x^a+b) / (1+x^b) = x^a
To determine if the given statements are true, let's analyze each statement separately.
1. (a/b)^x = a^x - b^x:
To verify the correctness of this equation, we need to perform the necessary algebraic operations and check if both sides of the equation are equivalent.
Starting with the left side:
(a/b)^x = (a/b)^x
To simplify further, we can write (a/b) as a fraction:
(a/b)^x = (a/b)^x = (a^x)/(b^x)
Now let's examine the right side:
a^x - b^x
Since the two sides should be equal, we can set them equal to each other:
(a^x)/(b^x) = a^x - b^x
To continue, we could multiply both sides of the equation by b^x to eliminate the fraction:
(a^x) = a^x * b^x - b^2x
However, at this point, we can observe that the equation in the original form does not hold true. By simplifying, we see that the statement (a/b)^x = a^x - b^x is NOT true.
2. (x^a + x^a+b) / (1+x^b) = x^a:
Similarly, let's take a closer look at this equation and determine its validity.
Begin with the left side:
(x^a + x^a+b) / (1+x^b)
To simplify further, we can factor out x^a from the numerator:
x^a * (1 + x^b-a) / (1 + x^b)
Now let's examine the right side:
x^a
To check if the equation holds true, we need to compare the two sides and confirm their equivalence:
x^a * (1 + x^b-a) / (1 + x^b) = x^a
Here, we can observe that the equation does hold true. By simplifying both sides, we can see that (x^a + x^a+b) / (1+x^b) = x^a.
In summary, the first statement (a/b)^x = a^x - b^x is NOT true, while the second statement (x^a + x^a+b) / (1+x^b) = x^a is indeed true.