square root of 700 simplified
The correct answer and explanation is:
The simplified form of the square root of 700 is 10710\sqrt{7}.
Here is how this simplification is done step-by-step:
- Start with the expression 700\sqrt{700}.
- Factor 700 into its prime factors or into a product where one part is a perfect square.
The number 700 can be written as 700=100×7700 = 100 \times 7. - Since 100 is a perfect square (because 100=102100 = 10^2), the square root of 100 is 10.
- Using the property of square roots a×b=a×b\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}, the square root of 700 can be split as: 700=100×7=100×7\sqrt{700} = \sqrt{100 \times 7} = \sqrt{100} \times \sqrt{7}
- This simplifies to: 10×710 \times \sqrt{7}
- Since 7\sqrt{7} cannot be simplified further (7 is a prime number), the expression 10710 \sqrt{7} is the simplest radical form.
Explanation of the process:
Square roots can often be simplified by identifying perfect square factors within the number under the root. Perfect squares are numbers like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so forth, because their square roots are whole numbers. When a number is broken down into the product of a perfect square and another factor, the square root of the perfect square can be taken out of the radical as a whole number.
In this case, 700 is factored into 100 and 7. The square root of 100 is 10, which can be taken out of the radical. The number 7 is not a perfect square, so it remains inside the square root. The expression becomes a multiplication of the number outside the square root and the remaining radical term.
This simplification is useful because it expresses the square root in a clearer form and makes it easier for further mathematical operations such as addition, subtraction, or comparison with other square roots. It also makes calculations easier when approximating, as 10710 \sqrt{7} can be approximated by calculating the value of 7\sqrt{7} and then multiplying by 10. Since 7\sqrt{7} is approximately 2.6457, 10×2.6457=26.45710 \times 2.6457 = 26.457, which is the decimal approximation of 700\sqrt{700}.
Thus, 10710\sqrt{7} is the simplest exact radical form of the square root of 700.