Question
Which number is equivalent to: 610−−√⋅39–√
?(1 point)
Responses
990−−√
9 square root of 90
919−−√
9 square root of 19
5410−−√
54 square root of 10
1809–√
?(1 point)
Responses
990−−√
9 square root of 90
919−−√
9 square root of 19
5410−−√
54 square root of 10
1809–√
Answers
GPT-4o mini
To simplify the expression \( \sqrt{610} \cdot \sqrt{39} \), we can first combine the square roots:
\[
\sqrt{610} \cdot \sqrt{39} = \sqrt{610 \cdot 39}
\]
Now let's calculate \( 610 \cdot 39 \):
1. **Multiply 610 by 39**:
\[
610 \cdot 39 = 610 \cdot (40 - 1) = 610 \cdot 40 - 610 \cdot 1 = 24400 - 610 = 23790
\]
So, we have:
\[
\sqrt{610} \cdot \sqrt{39} = \sqrt{23790}
\]
Next, we want to simplify \( \sqrt{23790} \). To do that, we first factor \( 23790 \):
2. **Prime factorization of \( 23790 \)**:
- \( 23790 \) is even, so divide by \( 2 \):
\[
23790 \div 2 = 11895
\]
- Now factor \( 11895 \). The sum of the digits \( 1 + 1 + 8 + 9 + 5 = 24 \) is divisible by \( 3 \).
\[
11895 \div 3 = 3965
\]
- Next, \( 3965 \) ends with \( 5 \), so divide by \( 5 \):
\[
3965 \div 5 = 793
\]
- Now, we check \( 793 \) for factors. It turns out \( 793 = 7 \times 113 \) (both are prime).
Thus, the full factorization of \( 23790 \) is:
\[
23790 = 2 \times 3 \times 5 \times 7 \times 113
\]
3. **Use the factorization to simplify \( \sqrt{23790} \)**:
- None of the prime factors are repeated, so no squares to take out. The square root remains:
\[
\sqrt{23790}
\]
This does not match any of the numerical options exactly. However, we know that:
\[
\sqrt{23790} \approx \sqrt{23790} \approx 154.175
\]
Assuming options were specifically defined values, let's check the numeric responses provided.
- We can approximate the square roots provided in responses for comparison:
1. \( \sqrt{990} \approx 31.4 \)
2. \( 9\sqrt{19} \approx 9 \times 4.3589 \approx 39.23 \)
3. \( 54\sqrt{10} \approx 54 \times 3.1623 \approx 171.34 \)
4. \( \sqrt{1809} \approx 42.5 \)
None of these options seem to relate closely to \( \sqrt{23790} \approx 154.175 \).
In conclusion, there might be an error in the options provided, or a misunderstanding. Based on calculation of \( \sqrt{610} \cdot \sqrt{39} = \sqrt{23790} \), this is the simplified form, which doesn't directly reduce to the numbers given in the options. Please double-check the options given to clarify if there's any additional detail.
\[
\sqrt{610} \cdot \sqrt{39} = \sqrt{610 \cdot 39}
\]
Now let's calculate \( 610 \cdot 39 \):
1. **Multiply 610 by 39**:
\[
610 \cdot 39 = 610 \cdot (40 - 1) = 610 \cdot 40 - 610 \cdot 1 = 24400 - 610 = 23790
\]
So, we have:
\[
\sqrt{610} \cdot \sqrt{39} = \sqrt{23790}
\]
Next, we want to simplify \( \sqrt{23790} \). To do that, we first factor \( 23790 \):
2. **Prime factorization of \( 23790 \)**:
- \( 23790 \) is even, so divide by \( 2 \):
\[
23790 \div 2 = 11895
\]
- Now factor \( 11895 \). The sum of the digits \( 1 + 1 + 8 + 9 + 5 = 24 \) is divisible by \( 3 \).
\[
11895 \div 3 = 3965
\]
- Next, \( 3965 \) ends with \( 5 \), so divide by \( 5 \):
\[
3965 \div 5 = 793
\]
- Now, we check \( 793 \) for factors. It turns out \( 793 = 7 \times 113 \) (both are prime).
Thus, the full factorization of \( 23790 \) is:
\[
23790 = 2 \times 3 \times 5 \times 7 \times 113
\]
3. **Use the factorization to simplify \( \sqrt{23790} \)**:
- None of the prime factors are repeated, so no squares to take out. The square root remains:
\[
\sqrt{23790}
\]
This does not match any of the numerical options exactly. However, we know that:
\[
\sqrt{23790} \approx \sqrt{23790} \approx 154.175
\]
Assuming options were specifically defined values, let's check the numeric responses provided.
- We can approximate the square roots provided in responses for comparison:
1. \( \sqrt{990} \approx 31.4 \)
2. \( 9\sqrt{19} \approx 9 \times 4.3589 \approx 39.23 \)
3. \( 54\sqrt{10} \approx 54 \times 3.1623 \approx 171.34 \)
4. \( \sqrt{1809} \approx 42.5 \)
None of these options seem to relate closely to \( \sqrt{23790} \approx 154.175 \).
In conclusion, there might be an error in the options provided, or a misunderstanding. Based on calculation of \( \sqrt{610} \cdot \sqrt{39} = \sqrt{23790} \), this is the simplified form, which doesn't directly reduce to the numbers given in the options. Please double-check the options given to clarify if there's any additional detail.