problem 1 simplifying square root expressions
got it? what is the simplified form of $\sqrt{64}$?
3. circle the equation that uses the positive square root of 64.
$16 \cdot 4 = 64$ $32 \cdot 2 = 64$ $8 \cdot 8 = 64$
4. the simplified form of $\sqrt{64}$ is .

problem 2 estimating a square root
got it? what is the value of $\sqrt{34}$ to the nearest integer?
5. use the number lines below to find the perfect squares closest to 34.
write 25, 34, and 36 in the correct positions on the number line.
complete the number line with square roots.
number line with 20, 30, 40 and $\sqrt{20}$, $\sqrt{30}$, $\sqrt{40}$, and a 5 marked
6. since 34 is closer to than to ,
$\sqrt{34}$ is closer to than to .
so, the value of $\sqrt{34}$ to the nearest integer is .

you can classify numbers using sets. a set is a well - defined collection of objects. each object in the set is called an element of the set. a subset of a set consists of elements from the given set. you can list the elements of a set within braces \{ \}.
7. complete the sets of numbers.
natural numbers $\left\{1, \quad, 3, \ldots\
ight\}$
whole numbers $\left\{ \quad, 1, \quad, 3, \ldots\
ight\}$
integers $\left\{\ldots, - 2, \quad, 0, 1, \quad, 3, \ldots\
ight\}

a rational number is any number that you can write in the form \(\frac{a}{b}$, where $a$ and $b$ are integers and $b \
eq 0$. a rational number in decimal form is either a terminating decimal such as 5.45 or a repeating decimal such as 0.333..., which you can write as $0.\overline{3}$.
8. cross out the numbers that are not rational numbers.
$\pi$ $\quad - \frac{7}{4}$ $\quad \sqrt{5}$ $\quad 0.\overline{9}$ $\quad 7.35$

an irrational number cannot be represented as the quotient of two integers. in decimal form, irrational numbers do not terminate or repeat. irrational numbers include $\pi$ and $\sqrt{2}$.

Question

problem 1 simplifying square root expressions
got it? what is the simplified form of $\sqrt{64}$?
3. circle the equation that uses the positive square root of 64.
$16 \cdot 4 = 64$ $32 \cdot 2 = 64$ $8 \cdot 8 = 64$
4. the simplified form of $\sqrt{64}$ is .

problem 2 estimating a square root
got it? what is the value of $\sqrt{34}$ to the nearest integer?
5. use the number lines below to find the perfect squares closest to 34.
write 25, 34, and 36 in the correct positions on the number line.
complete the number line with square roots.
number line with 20, 30, 40 and $\sqrt{20}$, $\sqrt{30}$, $\sqrt{40}$, and a 5 marked
6. since 34 is closer to  than to ,
$\sqrt{34}$ is closer to  than to .
so, the value of $\sqrt{34}$ to the nearest integer is .

you can classify numbers using sets. a set is a well - defined collection of objects. each object in the set is called an element of the set. a subset of a set consists of elements from the given set. you can list the elements of a set within braces \{ \}.
7. complete the sets of numbers.
natural numbers $\left\{1, \quad, 3, \ldots\
ight\}$
whole numbers $\left\{ \quad, 1, \quad, 3, \ldots\
ight\}$
integers $\left\{\ldots, - 2, \quad, 0, 1, \quad, 3, \ldots\
ight\}

a rational number is any number that you can write in the form \(\frac{a}{b}$, where $a$ and $b$ are integers and $b \
eq 0$. a rational number in decimal form is either a terminating decimal such as 5.45 or a repeating decimal such as 0.333..., which you can write as $0.\overline{3}$.
8. cross out the numbers that are not rational numbers.
$\pi$ $\quad - \frac{7}{4}$ $\quad \sqrt{5}$ $\quad 0.\overline{9}$ $\quad 7.35$

an irrational number cannot be represented as the quotient of two integers. in decimal form, irrational numbers do not terminate or repeat. irrational numbers include $\pi$ and $\sqrt{2}$.

Accepted Answer

### Problem 1 (Simplifying Square Root Expressions)
#### 3. Circle the equation that uses the positive square root of 64.
To find the positive square root of 64, we need to find a number $ x $ such that $ x \times x = 64 $ (since the square root of a number $ a $ is a number $ x $ where $ x^2 = a $, and the positive square root is the positive $ x $).
- For $ 16 \cdot 4 = 64 $: $ 16 $ and $ 4 $ are different, so this is not a square of a single number.
- For $ 32 \cdot 2 = 64 $: $ 32 $ and $ 2 $ are different, so this is not a square of a single number.
- For $ 8 \cdot 8 = 64 $: Here, $ 8 \times 8 = 64 $, so this represents the positive square root (since $ 8 $ is positive) of $ 64 $ because $ \sqrt{64}=8 $ and $ 8^2 = 64 $.

So we circle $ 8 \cdot 8 = 64 $.

#### 4. The simplified form of $ \sqrt{64} $ is:
We know that the square root of a number $ a $ is a number $ x $ such that $ x^2=a $. We need to find $ x $ where $ x^2 = 64 $.
We test numbers: $ 8^2=64 $ (since $ 8\times8 = 64 $) and also $ (-8)^2 = 64 $, but the square root symbol $ \sqrt{} $ denotes the principal (positive) square root. So $ \sqrt{64}=8 $.

### Problem 2 (Estimating a Square Root)
#### 5. Use the number lines below to find the perfect squares closest to 34.
Perfect squares are numbers like $ n^2 $ where $ n $ is an integer. Let's list some perfect squares:
- $ 5^2=25 $
- $ 6^2 = 36 $
- $ 4^2=16 $, $ 5^2 = 25 $, $ 6^2=36 $, $ 7^2 = 49 $

We need to find the perfect squares closest to 34. $ 25=5^2 $ and $ 36 = 6^2 $. Now, $ 34 - 25=9 $ and $ 36 - 34 = 2 $. Wait, no, wait: Wait, $ 25 $ is $ 5^2 $, $ 36 $ is $ 6^2 $. So on the number line for numbers: 20, 25, 30, 34, 36, 40. For the square root number line: $ \sqrt{20} $, $ \sqrt{25}=5 $, $ \sqrt{30} $, $ \sqrt{34} $, $ \sqrt{36}=6 $, $ \sqrt{40} $.

So we fill in the number line:
- For the number line (top): The positions are 20, 25, 30, 34, 36, 40 (since 25 is $ 5^2 $, 36 is $ 6^2 $, 34 is between them).
- For the square root number line (bottom): $ \sqrt{20} $, $ \sqrt{25} = 5 $, $ \sqrt{30} $, $ \sqrt{34} $, $ \sqrt{36}=6 $, $ \sqrt{40} $

#### 6. Since 34 is closer to 36 than to 25, $ \sqrt{34} $ is closer to 6 than to 5. So, the value of $ \sqrt{34} $ to the nearest integer is 6.
Let's check the distances:
- Distance between 34 and 25: $ 34 - 25=9 $
- Distance between 36 and 34: $ 36 - 34 = 2 $
Since 34 is closer to 36, $ \sqrt{34} $ is closer to $ \sqrt{36}=6 $ than to $ \sqrt{25}=5 $. So the nearest integer is 6.

#### 7. Complete the sets of numbers.
- **Natural numbers**: Natural numbers are positive integers starting from 1. So the set is $ \{1, 2, 3, \dots\} $ (because natural numbers are 1, 2, 3, ...).
- **Whole numbers**: Whole numbers include 0 and positive integers. So the set is $ \{0, 1, 2, 3, \dots\} $ (0 is included, then 1, 2, 3, ...).
- **Integers**: Integers include negative integers, zero, and positive integers. So the set is $ \{\dots, -2, -1, 0, 1, 2, 3, \dots\} $ (negative integers like -2, -1, then 0, 1, 2, 3, ...).

#### 8. Cross out the numbers that are NOT rational numbers.
A rational number can be written as $ \frac{a}{b} $ where $ a,b $ are integers and $ b
eq0 $.
- $ \pi $: $ \pi $ is an irrational number (it has a non - terminating, non - repeating decimal expansion and cannot be written as a fraction of two integers).
- $ -\frac{7}{4} $: This is a fraction of two integers ($ a=-7 $, $ b = 4 $, $ b
eq0 $), so it is rational.
- $ \sqrt{5} $: $ \sqrt{5}\approx2.2360679775\cdots $ is a non - terminating, non - repeating decimal and cannot be written as a fraction of two integers, so it is irrational.
- $ 0.\overline{9} $: $ 0.\overline{9}=0.999\cdots $ which is equal to 1 (a rational number, since $ 1=\frac{1}{1} $), or we can show $ x = 0.999\cdots $, $ 10x=9.999\cdots $, $ 10x - x=9 $, $ 9x = 9 $, $ x = 1 $), so it is rational.
- $ 7.35 $: $ 7.35=\frac{735}{100}=\frac{147}{20} $, which is a fraction of two integers, so it is rational.

So we cross out $ \pi $ and $ \sqrt{5} $.

### Answers:
3. Circle $ \boldsymbol{8\cdot8 = 64} $
4. $ \boldsymbol{8} $
5. Number line (top): 20, 25, 30, 34, 36, 40; Square root number line (bottom): $ \sqrt{20} $, $ \sqrt{25}=5 $, $ \sqrt{30} $, $ \sqrt{34} $, $ \sqrt{36}=6 $, $ \sqrt{40} $
6. 34 is closer to 36 than to 25; $ \sqrt{34} $ is closer to 6 than to 5; Nearest integer: $ \boldsymbol{6} $
7. Natural numbers: $ \{1, \boldsymbol{2}, 3, \dots\} $; Whole numbers: $ \{\boldsymbol{0}, 1, \boldsymbol{2}, 3, \dots\} $; Integers: $ \{\dots, -2, \boldsymbol{-1}, 0, 1, \boldsymbol{2}, 3, \dots\} $
8. Cross out $ \boldsymbol{\pi} $ and $ \boldsymbol{\sqrt{5}} $

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QUESTION IMAGE

Question

Explanation:

Problem 1 (Simplifying Square Root Expressions)

3. Circle the equation that uses the positive square root of 64.

4. The simplified form of \( \sqrt{64} \) is:

Problem 2 (Estimating a Square Root)

5. Use the number lines below to find the perfect squares closest to 34.

6. Since 34 is closer to 36 than to 25, \( \sqrt{34} \) is closer to 6 than to 5. So, the value of \( \sqrt{34} \) to the nearest integer is 6.

7. Complete the sets of numbers.

8. Cross out the numbers that are NOT rational numbers.

Answer: