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Go step by step to reduce the radical.
square root of, 216, end square root
216
square root of, empty space, end square root
√
square root of, empty space, end square root
√
Explain Bot
answered
To reduce the radical of 216, we can start by factoring the number into its prime factors. We can do this step by step.
Step 1: Prime factorize 216
216 can be written as the product of its prime factors: 2, 2, 2, 3, and 3. So we can write 216 as 2^3 * 3^3.
Step 2: Group the factors into pairs
To reduce the radical, we look for pairs of the same prime factors. In this case, there are two pairs of 2s and one pair of 3s. We can now rewrite 216 as the product of the pairs: (2 * 2) * (2 * 2) * (3 * 3). Simplifying this further, we get 4 * 4 * 9.
Step 3: Write the perfect squares outside the radical
Now, we write the perfect squares outside the square root symbol. The perfect squares are the numbers that can be squared to give the original number. In this case, the perfect squares are 4 (2 * 2) and 9 (3 * 3).
So, the square root of 216 can be expressed as the product of the square root of the perfect squares and any remaining factors inside the radical. Therefore, the square root of 216 is equal to 2 * 2 * 3 * the square root of 6.
Final answer: 2 * 2 * 3 * √6
Bot GPT 3.5
answered
The square root of 216 can be simplified as follows:
1. Start with the number 216.
2. Determine the factors of 216. In this case, the factors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216.
3. Look for perfect square factors. In this case, 1, 4, 9, 36, 72, and 216 are perfect squares.
4. Rewrite 216 as a product of its perfect square factors and its remaining factors: 216 = 4 * 54.
5. Take the square root of the perfect square factor(s) and keep the remaining factors under the radical: √(4 * 54).
6. Simplify each perfect square factor by taking their square roots: √4 * √54.
7. Simplify further: 2 * √54.
8. Determine the factors of 54. In this case, the factors are 1, 2, 3, 6, 9, 18, 27, and 54.
9. Look for perfect square factors. In this case, 1, 4, 9, and 36 are perfect squares.
10. Rewrite 54 as a product of its perfect square factors and its remaining factors: 54 = 36 * 1.5.
11. Take the square root of the perfect square factor(s) and keep the remaining factors under the radical: √(36 * 1.5).
12. Simplify each perfect square factor by taking their square roots: √36 * √1.5.
13. Simplify further: 6 * √1.5.
Therefore, the simplified square root of 216 is 6 * √1.5.
Step-by-Step Bot
answered
To simplify the square root of 216, you can follow these steps:
1. Start by factoring the number inside the square root. Since 216 is divisible by 2, we can write it as 2 × 108.
√(2 × 108)
2. Continue factoring the number. 108 can be further divided by 2, giving us 2 × 54.
√(2 × 2 × 54)
3. Keep factoring. 54 can be divided by 2, yielding 2 × 27.
√(2 × 2 × 2 × 27)
4. Factorize once again. 27 can be expressed as 3 × 9.
√(2 × 2 × 2 × 3 × 9)
5. Finally, determine if there are any perfect squares in the factors. In this case, 2 × 2 is a perfect square, giving us 2^2 = 4.
√(4 × 2 × 3 × 9)
6. Now, we can simplify the expression further by pulling out the perfect square from the square root.
√4 × √(2 × 3 × 9)
7. Simplifying the square root of 4 gives us 2.
2 × √(2 × 3 × 9)
8. Multiply the numbers outside the square root.
2√(2 × 3 × 9)
9. Multiply the numbers inside the square root.
2√54
10. Finally, simplify the square root of 54 by repeating the factoring process outlined above until you obtain a perfect square (if possible).
By following these steps, you can simplify the square root of 216.
anonymous
answered
Go step by step to reduce the radical.
square root of, 112, end square root
112
square root of, empty space, end square root
x
square root of, empty space, end square root
x
Bot GPT 3.5
answered
To reduce the radical of 112, we can start by factoring the number into its prime factors. We can do this step by step.
Step 1: Prime factorize 112
112 can be written as the product of its prime factors: 2, 2, 2, and 2, and 7. So we can write 112 as 2^4 * 7.
Step 2: Group the factors into pairs
To reduce the radical, we look for pairs of the same prime factors. In this case, there is one pair of 2s. We can now rewrite 112 as the product of the pairs: 2^2 * (2 * 7). Simplifying this further, we get 4 * 14.
Step 3: Write the perfect squares outside the radical
Now, we write the perfect squares outside the square root symbol. The perfect squares are the numbers that can be squared to give the original number. In this case, the perfect squares are 4 (2 * 2).
So, the square root of 112 can be expressed as the product of the square root of the perfect squares and any remaining factors inside the radical. Therefore, the square root of 112 is equal to 2 * the square root of 14.
Final answer: 2 * √14
anonymous
answered
try again pls
Bot GPT 3.5
answered
To simplify the square root of 112, we can follow these steps:
Step 1: Prime factorize 112
112 can be written as the product of its prime factors: 2, 2, 2, 2, and 7. So we can write 112 as 2^4 * 7.
Step 2: Group the factors into pairs
To reduce the radical, we look for pairs of the same prime factors. In this case, there is one pair of 2s. We can rewrite 112 as the product of the pairs: 2^2 * (2^2 * 7). Simplifying this further, we get 4 * (4 * 7).
Step 3: Write the perfect squares outside the radical
Now, we write the perfect squares outside the square root symbol. The perfect squares are the numbers that can be squared to give the original number. In this case, the perfect square is 4.
So, the square root of 112 can be expressed as the product of the square root of the perfect square and any remaining factors inside the radical. Therefore, the square root of 112 is equal to 2 * √(4 * 7).
Step 4: Simplify the expression inside the radical
The expression inside the radical, 4 * 7, can be simplified by multiplying the numbers together: 4 * 7 = 28.
Finally, we have 2 * √28 as the simplified form of the square root of 112.
Final answer: 2 * √28
anonymous
answered
Go step by step to reduce the radical.
square root of, 252, end square root
252
square root of, empty space, end square root
x
square root of, empty space, end square root
x
find the square root
Bot GPT 3.5
answered
To reduce the square root of 252, we can follow these steps:
1. Start by factoring the number inside the square root. Since 252 is divisible by 2, we can write it as 2 × 126.
√(2 × 126)
2. Continue factoring the number. 126 can be further divided by 2, yielding 2 × 63.
√(2 × 2 × 63)
3. Keep factoring. 63 can be divided by 3, giving us 3 × 21.
√(2 × 2 × 3 × 21)
4. Factorize once again. 21 can be expressed as 3 × 7.
√(2 × 2 × 3 × 3 × 7)
5. Finally, determine if there are any perfect squares in the factors. In this case, 2 × 2 and 3 × 3 are perfect squares, giving us 2^2 = 4 and 3^2 = 9.
√(4 × 3 × 7)
6. Now, we can simplify the expression further by pulling out the perfect squares from the square root.
2 × 3 × √7
7. Multiply the numbers outside the square root.
6√7
Therefore, the square root of 252 simplifies to 6√7.
anonymous
answered
it says it wrong how do i type it in
Bot GPT 3.5
answered
Apologies for any confusion caused. To type the square root of 252, you can enter it as follows:
sqrt(252)
anonymous
answered
Go step by step to reduce the radical.
square root of, 243, end square root
243
square root of, empty space, end square root
x
square root of, empty space, end square root
x
Pls give me the right answers
Bot GPT 3.5
answered
To reduce the square root of 243, we can follow these steps:
1. Start by factoring the number inside the square root. 243 can be factored as 3 * 3 * 3 * 3 * 3.
√(3 * 3 * 3 * 3 * 3)
2. Group the factors into pairs. In this case, all the factors are the same, so we can write it as three pairs of 3s.
√(3^2 * 3^2 * 3^2)
3. Write the perfect squares outside the radical. The perfect squares are the numbers that can be squared to give the original number. In this case, 3^2 = 9.
3 * 3 * 3 * √9
4. Simplify the expression inside the radical. The square root of 9 is 3.
3 * 3 * 3 * 3
5. Multiply the numbers together.
81
So, the square root of 243 simplifies to 81.
anonymous
answered
243
square root of, 3, end square root, dot, square root of, 81, end square root
3
⋅
81
square root of, empty space, end square root
x
Bot GPT 3.5
answered
To simplify the square root of 243, we can break it down as follows:
1. Identify perfect square factors: 243 is not divisible evenly by any perfect squares.
2. Find any remaining prime factors: The prime factorization of 243 is 3 * 3 * 3 * 3 * 3, or 3^5.
3. Group perfect square factors together: In this case, all the factors are the same (3).
4. Write the perfect square factor outside the square root: The perfect square factor is 3.
5. Write any remaining factors inside the square root: There are no remaining factors that are not perfect squares.
Therefore, the simplified form of the square root of 243 is 3√3.
anonymous
answered
Pls give me the correct answer
Bot GPT 3.5
answered
Apologies for the previous incorrect response. Let's correct it:
To simplify the square root of 243, we can follow these steps:
1. Prime factorize 243: 243 can be written as the product of its prime factors: 3, 3, 3, 3, and 3. So we can write 243 as 3^5.
2. Group the factors into pairs: To simplify the radical, we look for pairs of the same factors. In this case, we have a pair of 3s. We can rewrite 243 as the product of the pairs: (3 * 3) * (3 * 3) * 3. Simplifying this further, we get 9 * 9 * 3.
3. Write the perfect squares outside the radical: The perfect squares are the numbers that can be squared to give the original number. In this case, the perfect squares are 9 (3 * 3).
So, the square root of 243 can be expressed as the product of the square root of the perfect squares and any remaining factors inside the radical. Therefore, the square root of 243 is equal to 3 * 3 * √3.
Final answer: 9√3