Question

What is the irrational square root 72−−√
rewritten as the product of an integer and another irrational square root?(1 point)
Responses

38–√
3 start root 8 end root

362–√
36 start root 2 end root

62–√
6 start root 2 end root

218−−√
Answer
What is 54−−√
simplified?(1 point)
Responses

36–√
3 start root 6 end root

227−−√
2 start root 27 end root

272–√
27 start root 2 end root

69–√
Answer
What is the irrational cube root −16−−−−√3
rewritten as a product of an integer and another irrational cube root?(1 point)
Responses

22–√3
2 root index 3 start root 2 end root

−4
negative 4

42–√3
4 root index 3 start root 2 end root

−22–√3
Answer
Find the square root of 343.(1 point)
Responses

77–√
7 start root 7 end root

7–√
start root 7 end root

77–√3
7 root index 3 start root 7 end root

49
Answered by GPT-4o mini
To rewrite the irrational square root \( \sqrt{72} \) as the product of an integer and another irrational square root, we start by simplifying \( \sqrt{72} \).

1. Factor 72 into its prime factors:
\[
72 = 36 \times 2
\]

2. Now find the square root:
\[
\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}
\]

3. Since \( \sqrt{36} = 6 \):
\[
\sqrt{72} = 6 \sqrt{2}
\]

Thus, the correct answer is \(\mathbf{6 \sqrt{2}}\).
Answered by GPT-4o mini
To simplify \( \sqrt{54} \), we can factor it as follows:

1. Factor 54 into its prime factors:
\[
54 = 9 \times 6 = 9 \times 3 \times 2
\]
or simply:
\[
54 = 27 \times 2
\]

2. Now, we can simplify \( \sqrt{54} \):
\[
\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3 \sqrt{6}
\]
or,
\[
\sqrt{54} = \sqrt{27 \times 2} = \sqrt{27} \times \sqrt{2} = 3\sqrt{3} \times \sqrt{2} = 3\sqrt{6}
\]

Thus, the simplified form of \( \sqrt{54} \) is \( \mathbf{3 \sqrt{6}} \).

Based on the options you provided, the correct response is:

**3 start root 6 end root**.
Answered by GPT-4o mini
To rewrite the irrational cube root \( -\sqrt[3]{16} \) as a product of an integer and another irrational cube root, we start simplifying \( \sqrt[3]{16} \).

1. Factor 16:
\[
16 = 8 \times 2
\]
where \( 8 \) is a perfect cube.

2. Now, apply the cube root:
\[
\sqrt[3]{16} = \sqrt[3]{8 \times 2} = \sqrt[3]{8} \times \sqrt[3]{2} = 2 \sqrt[3]{2}
\]

Thus, we have:
\[
-\sqrt[3]{16} = - (2 \sqrt[3]{2}) = -2 \sqrt[3]{2}
\]

Therefore, the expression \( -\sqrt[3]{16} \) can be rewritten as:
**−2 root index 3 start root 2 end root**.
Answered by GPT-4o mini
To find the square root of \( 343 \), we first factor \( 343 \):

1. Note that \( 343 = 7 \times 7 \times 7 = 7^3 \).

2. We can express \( 343 \) in the form of a square root:
\[
\sqrt{343} = \sqrt{7^3} = \sqrt{7^2 \times 7} = 7 \sqrt{7}
\]

Thus, the square root of \( 343 \) is:
\[
\mathbf{7 \sqrt{7}}
\]

Therefore, the correct response is:
**7 start root 7 end root**.