Rational & Irrational Numbers

[nycmath]

Precalculus
David Cohen
Edition 3
Chapter 1, Section 1.1

65 (a) Give an example in which the result of raising a rational number to a rational power is an irrational number.

65 (b) Give an example in which the result of raising an irrational number to a rational power is a rational numer.

[Math Tutor]

(a) Base 2, exponent 1/2:

[tex]2^{\frac{1}{2}}=\sqrt{2}[/tex]

Rational base, rational exponent, irrational answer. Any non-perfect-square works.

(b) Base [tex]\sqrt{2}[/tex], exponent 2:

[tex]\left(\sqrt{2}\right)^{2}=2[/tex]

Irrational base, rational exponent, rational answer.

Squaring undoes the square root, so (b) is just (a) run backwards.

[nycmath]

(a) Base 2, exponent 1/2:

[tex]2^{\frac{1}{2}}=\sqrt{2}[/tex]

Rational base, rational exponent, irrational answer. Any non-perfect-square works.

(b) Base [tex]\sqrt{2}[/tex], exponent 2:

[tex]\left(\sqrt{2}\right)^{2}=2[/tex]

Irrational base, rational exponent, rational answer.

Squaring undoes the square root, so (b) is just (a) run backwards.

I found this set of questions to be interesting.