Question

for which value of p and w is p + w a rational number?

1. $p = \frac{1}{\sqrt{3}}$ and $w = \frac{1}{\sqrt{6}}$
2. $p = \frac{1}{\sqrt{4}}$ and $w = \frac{1}{\sqrt{9}}$
3. $p = \frac{1}{\sqrt{6}}$ and $w = \frac{1}{\sqrt{10}}$
4. $p = \frac{1}{\sqrt{25}}$ and $w = \frac{1}{\sqrt{2}}$

1.
given: $l = \sqrt{2}$
$m = 3\sqrt{3}$
$n = \sqrt{16}$
$p = \sqrt{9}$
which expression results in a rational number?

1. $l + m$
2. $m + n$
3. $n + p$
4. $p + l$

2.
the product of $\sqrt{576}$ and $\sqrt{684}$ is

1. irrational because both factors are irrational
2. rational because both factors are rational
3. irrational because one factor is irrational
4. rational because one factor is rational

3.
given the following expressions:
i. $-\frac{5}{8} + \frac{3}{5}$ iii. $\left(\sqrt{5}\
ight) \cdot \left(\sqrt{5}\
ight)$
ii. $\frac{1}{2} + \sqrt{2}$ iv. $3 \cdot \left(\sqrt{49}\
ight)$
which expression(s) result in an irrational number?

1. ii, only
2. iii, only
3. i, iii, iv
4. ii, iii, iv

Explanation:

Problem 1

Step1: Simplify each \( P \) and \( W \)

• Option 1: \( P = \frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3} \) (irrational), \( W = \frac{1}{\sqrt{6}}=\frac{\sqrt{6}}{6} \) (irrational). Sum: irrational + irrational, likely irrational.
• Option 2: \( P = \frac{1}{\sqrt{4}}=\frac{1}{2} \) (rational), \( W = \frac{1}{\sqrt{9}}=\frac{1}{3} \) (rational). Sum: \( \frac{1}{2}+\frac{1}{3}=\frac{5}{6} \) (rational).
• Option 3: \( P = \frac{1}{\sqrt{6}}=\frac{\sqrt{6}}{6} \) (irrational), \( W = \frac{1}{\sqrt{10}}=\frac{\sqrt{10}}{10} \) (irrational). Sum: irrational + irrational, likely irrational.
• Option 4: \( P = \frac{1}{\sqrt{25}}=\frac{1}{5} \) (rational), \( W = \frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2} \) (irrational). Sum: rational + irrational, irrational.

Step1: Simplify \( L, M, N, P \)

• \( L = \sqrt{2} \) (irrational), \( M = 3\sqrt{3} \) (irrational), \( N = \sqrt{16}=4 \) (rational), \( P = \sqrt{9}=3 \) (rational).

Step2: Check each option

• Option 1: \( L + M \): irrational + irrational, irrational.
• Option 2: \( M + N \): irrational + rational, irrational.
• Option 3: \( N + P \): \( 4 + 3 = 7 \) (rational).
• Option 4: \( P + L \): rational + irrational, irrational.

Step1: Simplify \( \sqrt{576} \) and \( \sqrt{684} \)

• \( \sqrt{576}=24 \) (rational), \( \sqrt{684}=\sqrt{4\times171}=\sqrt{4\times9\times19}=6\sqrt{19} \) (irrational).

Step2: Analyze product

• Product: \( 24\times6\sqrt{19}=144\sqrt{19} \) (irrational, since \( \sqrt{19} \) is irrational). But wait, wait—wait, \( \sqrt{576}=24 \) (rational), \( \sqrt{684} \) is irrational. So product is irrational because one factor ( \( \sqrt{684} \)) is irrational. Wait, no—wait, the options:
• Option 1: Wrong, \( \sqrt{576} \) is rational.
• Option 2: Wrong, \( \sqrt{684} \) is irrational.
• Option 3: Correct, irrational because one factor ( \( \sqrt{684} \)) is irrational.
• Option 4: Wrong, product is irrational, and reason is wrong.

Answer:

1. \( P = \frac{1}{\sqrt{4}} \) and \( W = \frac{1}{\sqrt{9}} \)

Problem 2