Determine which two values the following irrational number is between.

Question

Determine which two values the following irrational number is between.
square root of 21

A. 
square root of 21 is between 3.1 and 4.1
  B. 
 square root of 21 is between 4.1 and 5.1
  C. 
 square root of 21 is between 6.1 and 7.1
  D. 
 square root of 21 is between 5.1 and 6.1

GPT-4o mini · Answer

To determine between which two values the square root of 21 lies, we first need to find the perfect squares around 21.

The perfect squares closest to 21 are:

$4^2 = 16$
$5^2 = 25$

This means that: \[ 4 < \sqrt{21} < 5 \]

Next, we can square the numbers in the options to find which interval contains 21.

Calculating the squares of 4.1 and 5.1, we find:

$4.1^2 = 16.81$
$5.1^2 = 26.01$

Since $16.81 < 21 < 26.01$, this confirms that $4.1$ and $5.1$ are the boundaries.

Thus, the correct interval is:

B. square root of 21 is between 4.1 and 5.1.

Answer

Place a point on the number line to represent the approximate value of 4 square root of 12. Round to the nearest tenth 13.5——————14.0——————14.5

GPT-4o mini · Answer

To find the approximate value of $4 \sqrt{12}$, we start by simplifying $ \sqrt{12} $: $$ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3} $$ Now, we will multiply by 4: $$ 4 \sqrt{12} = 4 \times 2 \sqrt{3} = 8 \sqrt{3} $$ Next, we will find the approximate value of $ \sqrt{3} $. A common approximation is: $$ \sqrt{3} \approx 1.732 $$ Now, substitute this value into the equation: $$ 8 \sqrt{3} \approx 8 \times 1.732 = 13.856 $$ Now, we round $13.856$ to the nearest tenth, which gives us $13.9$. Considering the number line given: ``` 13.5 —————— 14.0 —————— 14.5 ``` The point $13.9$ is slightly before $14.0$. So, you would place a point just before the line labeled $14.0$ on the number line.

Answer

Identify the missing symbol. square root of 439 ? 7 pi A. = B. > C. <

GPT-4o mini · Answer

To determine the relationship between $\sqrt{439}$ and $7\pi$, we first need to approximate the value of both expressions. 1. **Calculating $\sqrt{439}$**: - Since $21^2 = 441$ and $20^2 = 400$, we know that $20 < \sqrt{439} < 21$. - To get a more accurate estimate, we can calculate: $$ \sqrt{439} \approx 20.9 \, (\text{since it's closer to } 441) $$ 2. **Calculating $7\pi$**: - Using the approximation $\pi \approx 3.14$: $$ 7\pi \approx 7 \times 3.14 = 21.98 $$ Now we compare the two values: - $\sqrt{439} \approx 20.9$ - $7\pi \approx 21.98$ Since $20.9 < 21.98$, we can conclude: $$ \sqrt{439} < 7\pi $$ Thus, the missing symbol is: **C. <**

Answer

At what position on the number line is the red dot located? A. square root of 44 B. square root of 34 C. square root of 88 D. square root of 54

GPT-4o mini · Answer

To determine the position of the red dot on the number line in relation to the given options, we can calculate the approximate values of each square root. Let's find the approximate decimal values for each square root.

Square root of 44: \[ \sqrt{44} \approx 6.63 \quad (\text{since } 6^2 = 36 \text{ and } 7^2 = 49) \]
Square root of 34: \[ \sqrt{34} \approx 5.83 \quad (\text{since } 5^2 = 25 \text{ and } 6^2 = 36) \]
Square root of 88: \[ \sqrt{88} \approx 9.38 \quad (\text{since } 9^2 = 81 \text{ and } 10^2 = 100) \]
Square root of 54: \[ \sqrt{54} \approx 7.35 \quad (\text{since } 7^2 = 49 \text{ and } 8^2 = 64) \]

Now we have the following approximate values:

A. $\sqrt{44} \approx 6.63$
B. $\sqrt{34} \approx 5.83$
C. $\sqrt{88} \approx 9.38$
D. $\sqrt{54} \approx 7.35$

If we need to identify the location of the red dot in relation to a number line, please provide how the number line is marked or any reference points, and then I would be able to help you determine where the red dot is located based on the approximate values of these square roots.

Answer

5————6——-•——7————8

GPT-4o mini · Answer

With the number line marked as: ``` 5 ———— 6 ———- • ——— 7 ————— 8 ``` we can evaluate the approximate values of the square roots we calculated earlier: 1. **A. $\sqrt{44} \approx 6.63$** - This value is between $6$ and $7$. Specifically, it is closer to $7$. 2. **B. $\sqrt{34} \approx 5.83$** - This value is between $5$ and $6$. Specifically, it is closer to $6$. 3. **C. $\sqrt{88} \approx 9.38$** - This value is outside of the range of $5$ to $8$, on the number line you provided. 4. **D. $\sqrt{54} \approx 7.35$** - This value is also between $7$ and $8$. Specifically, it is closer to $7$. Given that the red dot is located between $6$ and $7$: - $\sqrt{34} \approx 5.83$ is the only value that corresponds closely with this point at position $•$. Based on the number line you provided, the red dot is likely located at: **B. $\sqrt{34}$**.

Answer

Which of the following is the best approximation for the irrational number shown below? – square root of 304 A. -17.45 B. -17.42 C. -17.43 D. -17.44

GPT-4o mini · Answer

To approximate $-\sqrt{304}$, we first need to find the value of $\sqrt{304}$. 1. **Calculating $\sqrt{304}$**: - We know that $17^2 = 289$ and $18^2 = 324$, so: $$ 17 < \sqrt{304} < 18 $$ - To get a more accurate approximation, we can use a calculator or further simplify: - $\sqrt{304} = \sqrt{16 \times 19} = 4\sqrt{19}$ Using the approximation $\sqrt{19} \approx 4.36$: $$ 4 \sqrt{19} \approx 4 \times 4.36 \approx 17.44 $$ So, $$ \sqrt{304} \approx 17.44 $$ 2. **Finding $-\sqrt{304}$**: - Therefore: $$ -\sqrt{304} \approx -17.44 $$ Now we can compare this result with the options given: A. $-17.45$ B. $-17.42$ C. $-17.43$ D. $-17.44$ The closest approximation to $-17.44$ is: **D. -17.44**.