Question

1. sortify true or false for each statement

a. √10 is between 3 and 4? ○true ○false
b. √20 is between 4 and 5? ○true ○false
c. 5 - √3 is between 3 and 4? ○true ○false
d. √15 is between 3 and 2? ○true ○false
e. √120 is between 12 and 11? ○true ○false

1. which expression is less than 4.07? circle all that apply.

a. √14
b. 4π
c. √15
d. 3√2
e. √24

1. is the number between 4 and 5? select yes or no.

a. 3√2 ○no ○yes
b. 2√6 ○no ○yes
c. (1/2)√68 ○no ○yes
d. √127 ○no ○yes
e. 2√7 ○no ○yes

1. look at each square root. which are irrational? circle all that apply.

a. √10
b. √12
c. √16
d. √36
e. √48
f. √24

Explanation:

Problem 9: Which expressions are less than \( 4.07 \)?

Step 1: Evaluate each option

• Option A: \( \sqrt{1.6} \)

Calculate \( \sqrt{1.6} \approx 1.26 \), which is less than \( 4.07 \).

• Option B: \( 4\pi \)

\( \pi \approx 3.14 \), so \( 4\pi \approx 12.56 \), which is greater than \( 4.07 \).

• Option C: \( \sqrt{16} \)

\( \sqrt{16} = 4 \), which is less than \( 4.07 \).

• Option D: \( 3\sqrt{2} \)

\( \sqrt{2} \approx 1.414 \), so \( 3\sqrt{2} \approx 4.24 \), which is greater than \( 4.07 \).

• Option E: \( \sqrt{24} \)

\( \sqrt{24} \approx 4.90 \), which is greater than \( 4.07 \).

Problem 10: Is the number between 4 and 5?

Step 1: Analyze each option

• Option A: \( 3\sqrt{2} \approx 4.24 \)

\( 4 < 4.24 < 5 \), so Yes.

• Option B: \( 2\sqrt{5} \approx 4.47 \)

\( 4 < 4.47 < 5 \), so Yes.

• Option C: \( \frac{1}{2}\sqrt{68} \approx \frac{1}{2}(8.25) \approx 4.12 \)

\( 4 < 4.12 < 5 \), so Yes.

• Option D: \( \sqrt{27} \approx 5.20 \)

\( 5.20 > 5 \), so No.

• Option E: \( 2\sqrt{7} \approx 5.29 \)

\( 5.29 > 5 \), so No.

Problem 11: Which square roots are irrational?

A square root is irrational if the radicand is not a perfect square.

• Option A: \( \sqrt{10} \)

10 is not a perfect square, so \( \sqrt{10} \) is irrational.

• Option B: \( \sqrt{12} \)

12 is not a perfect square, so \( \sqrt{12} \) is irrational.

• Option C: \( \sqrt{16} = 4 \)

16 is a perfect square, so \( \sqrt{16} \) is rational.

• Option D: \( \sqrt{36} = 6 \)

36 is a perfect square, so \( \sqrt{36} \) is rational.

• Option E: \( \sqrt{48} \)

48 is not a perfect square, so \( \sqrt{48} \) is irrational.

• Option F: \( \sqrt{24} \)

24 is not a perfect square, so \( \sqrt{24} \) is irrational.

Final Answers

Problem 9:

Circle: \( \boldsymbol{\sqrt{1.6}} \), \( \boldsymbol{\sqrt{16}} \)

Problem 10:

• A: \( \boldsymbol{\text{Yes}} \)
• B: \( \boldsymbol{\text{Yes}} \)
• C: \( \boldsymbol{\text{Yes}} \)
• D: \( \boldsymbol{\text{No}} \)
• E: \( \boldsymbol{\text{No}} \)

Problem 11:

Circle: \( \boldsymbol{\sqrt{10}} \), \( \boldsymbol{\sqrt{12}} \), \( \boldsymbol{\sqrt{48}} \), \( \boldsymbol{\sqrt{24}} \)

Answer:

Problem 9: Which expressions are less than \( 4.07 \)?

Step 1: Evaluate each option

• Option A: \( \sqrt{1.6} \)

Calculate \( \sqrt{1.6} \approx 1.26 \), which is less than \( 4.07 \).

• Option B: \( 4\pi \)

\( \pi \approx 3.14 \), so \( 4\pi \approx 12.56 \), which is greater than \( 4.07 \).

• Option C: \( \sqrt{16} \)

\( \sqrt{16} = 4 \), which is less than \( 4.07 \).

• Option D: \( 3\sqrt{2} \)

\( \sqrt{2} \approx 1.414 \), so \( 3\sqrt{2} \approx 4.24 \), which is greater than \( 4.07 \).

• Option E: \( \sqrt{24} \)

\( \sqrt{24} \approx 4.90 \), which is greater than \( 4.07 \).

Problem 10: Is the number between 4 and 5?

Step 1: Analyze each option

• Option A: \( 3\sqrt{2} \approx 4.24 \)

\( 4 < 4.24 < 5 \), so Yes.

• Option B: \( 2\sqrt{5} \approx 4.47 \)

\( 4 < 4.47 < 5 \), so Yes.

• Option C: \( \frac{1}{2}\sqrt{68} \approx \frac{1}{2}(8.25) \approx 4.12 \)

\( 4 < 4.12 < 5 \), so Yes.

• Option D: \( \sqrt{27} \approx 5.20 \)

\( 5.20 > 5 \), so No.

• Option E: \( 2\sqrt{7} \approx 5.29 \)

\( 5.29 > 5 \), so No.

Problem 11: Which square roots are irrational?

A square root is irrational if the radicand is not a perfect square.

• Option A: \( \sqrt{10} \)

10 is not a perfect square, so \( \sqrt{10} \) is irrational.

• Option B: \( \sqrt{12} \)

12 is not a perfect square, so \( \sqrt{12} \) is irrational.

• Option C: \( \sqrt{16} = 4 \)

16 is a perfect square, so \( \sqrt{16} \) is rational.

• Option D: \( \sqrt{36} = 6 \)

36 is a perfect square, so \( \sqrt{36} \) is rational.

• Option E: \( \sqrt{48} \)

48 is not a perfect square, so \( \sqrt{48} \) is irrational.

• Option F: \( \sqrt{24} \)

24 is not a perfect square, so \( \sqrt{24} \) is irrational.

Final Answers

Problem 9:

Circle: \( \boldsymbol{\sqrt{1.6}} \), \( \boldsymbol{\sqrt{16}} \)

Problem 10:

• A: \( \boldsymbol{\text{Yes}} \)
• B: \( \boldsymbol{\text{Yes}} \)
• C: \( \boldsymbol{\text{Yes}} \)
• D: \( \boldsymbol{\text{No}} \)
• E: \( \boldsymbol{\text{No}} \)

Problem 11:

Circle: \( \boldsymbol{\sqrt{10}} \), \( \boldsymbol{\sqrt{12}} \), \( \boldsymbol{\sqrt{48}} \), \( \boldsymbol{\sqrt{24}} \)