Final Answer:
The value of the expression (10 + 5/3 + 10^(-5)/3)^2 is a rational number.
Step-by-step explanation:
A rational number is any number that can be expressed as a fraction where the numerator and denominator are integers. To show that the given expression is rational, we need to demonstrate that it can be written as a fraction with integer numerator and denominator.
Simplify the expression: First, simplify the terms within the parentheses:
5/3 + 10^(-5)/3 = 5/3 + 0.00000000333 (approximately)
Therefore, the expression becomes:
(10 + 5/3 + 0.00000000333)^2
Expand the square: Next, square the entire expression:
(10 + 5/3 + 0.00000000333)^2 = (10 + 1.6666666666 + 0.00000000333)^2
Expanding the square gives us a polynomial with integer coefficients.
Therefore, rational: Since all the terms in the expanded polynomial have integer coefficients, the entire expression can be written as a fraction with an integer numerator and denominator. This makes the value of the expression a rational number.
In conclusion, despite the presence of decimals and fractions within the original expression, simplifying and squaring it reveals its true nature as a rational number.