Question
prove that the sum of 3/4 and 10squared is irrational, show your work and in 1 sentence expliain why the sum is irattional
Answers
GPT-4o mini
To determine whether the sum of \( \frac{3}{4} \) and \( 10^2 \) is irrational, we first need to calculate the values of these two components.
1. Calculate \( 10^2 \):
\[
10^2 = 100
\]
2. Now, add \( \frac{3}{4} \) and \( 100 \):
\[
\frac{3}{4} + 100 = \frac{3}{4} + \frac{100 \times 4}{4} = \frac{3}{4} + \frac{400}{4} = \frac{3 + 400}{4} = \frac{403}{4}
\]
3. The result \( \frac{403}{4} \) is a rational number since it can be expressed as the ratio of two integers (403 and 4).
Thus, the sum of \( \frac{3}{4} \) and \( 10^2 \) is rational, not irrational.
In one sentence: The sum \( \frac{403}{4} \) is rational because it can be expressed as the quotient of two integers.
1. Calculate \( 10^2 \):
\[
10^2 = 100
\]
2. Now, add \( \frac{3}{4} \) and \( 100 \):
\[
\frac{3}{4} + 100 = \frac{3}{4} + \frac{100 \times 4}{4} = \frac{3}{4} + \frac{400}{4} = \frac{3 + 400}{4} = \frac{403}{4}
\]
3. The result \( \frac{403}{4} \) is a rational number since it can be expressed as the ratio of two integers (403 and 4).
Thus, the sum of \( \frac{3}{4} \) and \( 10^2 \) is rational, not irrational.
In one sentence: The sum \( \frac{403}{4} \) is rational because it can be expressed as the quotient of two integers.