Question

Consider the expression 36 −−−√
x 5/3
. Will the product be rational or irrational? Explain how you know in one or two sentences, and then find the product. (2 points)

Answer 1

you have to answer in two sentences and then find the product

Answer 2

To determine whether the expression \( \sqrt{36} \times x^{5/3} \) is rational or irrational, we first simplify \( \sqrt{36} \), which equals \( 6 \) (a rational number). The expression then becomes \( 6 \times x^{5/3} \).

The product will be rational if \( x^{5/3} \) is a rational number, which depends on whether \( x \) itself is rational. If \( x \) is rational, then \( 6 \times x^{5/3} \) will also be rational; if \( x \) is irrational, then \( 6 \times x^{5/3} \) will be irrational.

Assuming \( x \) is a rational number, we can make it explicit:

1. Let \( x = q \) where \( q \) is rational.
2. Then the product is \( 6 \times q^{5/3} \).

Thus, without loss of generality, we can find the product as follows, assuming \( x \) is a rational number. If \( x = 1 \), for simplicity, then:

\[
6 \times 1^{5/3} = 6 \times 1 = 6.
\]

Thus, the product will be rational, and examples can vary according to the value of \( x \). If \( x \) is irrational, the expression would yield an irrational product.

Answer 3

The expression \( \sqrt{36} \times x^{5/3} \) will be rational if \( x \) is rational, since \( \sqrt{36} = 6 \) is rational. If \( x = 1 \), the product is \( 6 \times 1^{5/3} = 6 \).