The cube root of 125 is 5.
To calculate this, you can raise 5 to the power of 3:
5³ = 5 × 5 × 5 = 125
The square root of 125 can be written in radical form as:
√125 = √(5^2 × 5) = 5√5
To calculate the square root of 125, you can use a calculator or do it manually. Here are both options:
**Using a Calculator:**
If you have a calculator, simply type in "sqrt(125)" and press enter. The answer will be displayed as approximately 11.18.
**Manual Calculation:**
To calculate the square root of 125 manually, you can use the following steps:
1. Find the perfect square that is closest to 125. In this case, it's 121 (which is the square of 11).
2. Since 125 is greater than 121, you know that the square root of 125 must be slightly larger than 11.
3. To estimate the correct value, you can try squaring a number close to 11 to see if it gets you closer to 125.
For example:
* 11^2 = 121 (too low)
* 12^2 = 144 (too high)
* 11.1^2 = 123.21 (close enough!)
So, the square root of 125 is approximately 11.18.
Note: Calculators and computers use advanced algorithms to calculate square roots quickly and accurately. If you need a more precise answer or want to double-check your calculation, using a calculator is always the best option!
To simplify the expression 3√125, we need to find the prime factorization of 125 and then take out any perfect squares.
The prime factorization of 125 is:
125 = 5 × 5 × 5
So, we can write √125 as:
√125 = √(5 × 5 × 5) = 5√25
Now, since 25 is a perfect square (it's the square of 5), we can simplify it further:
5√25 = 5 × 5 = 25
So, 3√125 simplifies to:
3 × 25 = 75
The square of 125 is:
125 × 125 = 15,625
The square root of 64 is... (drumroll please)... 8!
That's because 8 multiplied by 8 equals 64. In mathematical notation, that would be:
√64 = 8
To find the square root of 125 using the division method, you can follow these steps:
1. Write the number you want to find the square root of: 125
2. Think of a perfect square that is less than or equal to 125. A good starting point would be 9 (since 3^2 = 9), but we'll try something slightly larger, like 11.
3. Divide 125 by your chosen number (in this case, 11): 125 ÷ 11 = 11 remainder 4
4. Since the remainder is not zero, you know that 11 is not the correct square root. However, you can use the remainder to estimate the next possible perfect square.
5. Try dividing 125 by a number close to 11 (but slightly larger), like 12: 125 ÷ 12 = 10 remainder 5
6. Again, since the remainder is not zero, 12 is not the correct square root. But you can use the remainder to estimate the next possible perfect square.
7. Keep dividing and adjusting your guess until you find a number that divides evenly into 125 with no remainder.
Let's try again:
1. Divide 125 by 13: 125 ÷ 13 = 9 remainder 8
2. Since the remainder is not zero, 13 is not the correct square root.
3. Try dividing 125 by 14: 125 ÷ 14 = 8 remainder 11
4. Again, since the remainder is not zero, 14 is not the correct square root.
5. Keep going:
1. Divide 125 by 15: 125 ÷ 15 = 8 remainder 5
2. Since the remainder is not zero, 15 is not the correct square root.
3. Try dividing 125 by 16: 125 ÷ 16 = 7 remainder 13
4. Again, since the remainder is not zero, 16 is not the correct square root.
5. Keep going:
1. Divide 125 by 17: 125 ÷ 17 = 7 remainder 4
2. Since the remainder is small, you can try dividing again with a slightly larger number:
3. Divide 125 by 18: 125 ÷ 18 = 6 remainder 13 (not good enough)
4. Try again with an even larger number:
5. Divide 125 by 19: 125 ÷ 19 = 6 remainder 7 (getting closer!)
6. Finally, try dividing again with a slightly larger number:
7. Divide 125 by 20: 125 ÷ 20 = 6 remainder 5 (almost there!)
Since the remainder is small, you can stop here and conclude that:
√125 ≈ 11
Note that this method may take some trial and error, but it can be a useful way to estimate square roots when you don't have a calculator handy.
The square root of 100 is 10.
√100 = 10