Question

1. given the following expressions circle the ones that are irrational.

√2⋅√32 √4 + π 1/2 + √(9/4)
(2√3)^2 (2√3)^3 1/3
explain why these expressions are irrational.

Explanation:

Step1: Simplify each expression

• $\sqrt{2}\cdot\sqrt{32}=\sqrt{2\times32}=\sqrt{64} = 8$, which is rational.
• $\sqrt{4}+\pi=2 + \pi$. Since $\pi$ is irrational and the sum of a rational number ($2$) and an irrational number ($\pi$) is irrational.
• $\frac{1}{2}+\sqrt{\frac{9}{4}}=\frac{1}{2}+\frac{3}{2}=\frac{1 + 3}{2}=2$, which is rational.
• $(2\sqrt{3})^2=2^2\times(\sqrt{3})^2=4\times3 = 12$, which is rational.
• $(2\sqrt{3})^3=2^3\times(\sqrt{3})^3=8\times3\sqrt{3}=24\sqrt{3}$. Since $\sqrt{3}$ is irrational, $24\sqrt{3}$ is irrational.
• $\frac{1}{3}$ is a fraction of two integers, so it is rational.

Answer:

The irrational expressions are $\sqrt{4}+\pi$ and $(2\sqrt{3})^3$.