Question

1. while solving the equation 3(x - 2) = 6, becca wrote 3x - 6 = 6. which property did she use?
2. distributive
3. associative
4. commutative
5. identity
6. when solving the equation 4x² - 5 = 25, laura wrote 4x² = 30, as her first step. which property justifies lauras first step?
7. distributive property of multiplication over addition
8. multiplication property of equality
9. commutative property of addition
10. addition property of equality
11. 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: Analyze problem 6

Becca went from $3(x - 2)$ to $3x-6$. The distributive property $a(b - c)=ab - ac$ was used where $a = 3$, $b=x$ and $c = 2$.

Step2: Analyze problem 7

Laura went from $4x^{2}-5 = 25$ to $4x^{2}=30$. She added 5 to both sides of the equation. The addition property of equality states that if $a=b$, then $a + c=b + c$. Here $a = 4x^{2}-5$, $b = 25$ and $c = 5$.

Step3: Analyze problem 8 - Simplify expressions

• For $\sqrt{2}\cdot\sqrt{32}=\sqrt{2\times32}=\sqrt{64}=8$ (rational).
• For $\sqrt{4}+\pi=2 + \pi$. Since $\pi$ is irrational, $2+\pi$ is irrational.
• For $\frac{1}{2}+\sqrt{\frac{9}{4}}=\frac{1}{2}+\frac{3}{2}=2$ (rational).
• For $(2\sqrt{3})^{2}=2^{2}\times(\sqrt{3})^{2}=4\times3 = 12$ (rational).
• For $(2\sqrt{3})^{3}=2^{3}\times(\sqrt{3})^{3}=8\times3\sqrt{3}=24\sqrt{3}$. Since $\sqrt{3}$ is irrational, $24\sqrt{3}$ is irrational.
• For $\frac{1}{3}$ (rational).

$\sqrt{4}+\pi$ is irrational because $\pi$ is irrational and the sum of a rational number ($\sqrt{4}=2$) and an irrational number is irrational. $(2\sqrt{3})^{3}=24\sqrt{3}$ is irrational because $\sqrt{3}$ is irrational and multiplying a non - zero rational number ($24$) by an irrational number gives an irrational number.

Answer:

1. 1) distributive
2. 4) addition property of equality
3. Irrational expressions: $\sqrt{4}+\pi$, $(2\sqrt{3})^{3}$