Question

Mary makes the claim that when 2^(p/q), where p and q are both integers and q > 0, is multiplied by 2^(n), the product will always be rational. Is Mary's claim correct? Justify your answer.

a) Mary's claim is correct.
b) Mary's claim is incorrect.

Answer 1

Mary's claim is correct. The product of 2 raised to a rational power and 2 raised to an integer power will always be a rational number, as the rules of exponents dictate that these powers will add to form another rational exponent.

Step-by-step explanation:

Mary's claim that when 2p/q, where p and q are both integers and q > 0, is multiplied by 2n, the product will always be rational. To determine if Mary's claim is correct, we must understand the properties of exponents and rational numbers.

Rational numbers are numbers that can be expressed as the quotient or fraction p/q of two integers, where p is the numerator and q (the denominator) is not zero. Since p and q are integers in the expression 2p/q, this represents a rational number to an integer power when we multiply by another power of two, we're using the exponent rule for multiplying powers with the same base:

2p/q × 2n = 2(p/q) + n

The sum of a rational number and an integer is always rational, meaning the resulting exponent is rational, and thus the product is a rational number.

Therefore, Mary's claim is correct.