Find the two square roots for 81
The correct Answer and Explanation is:
The two square roots of 81 are:
- Positive square root: ( \sqrt{81} = 9 )
- Negative square root: ( -\sqrt{81} = -9 )
Explanation:
A square root of a number ( x ) is a value that, when multiplied by itself, equals ( x ). The number 81 can be written as ( 81 = 9 \times 9 ). Therefore, 9 is one of the square roots of 81, because ( 9 \times 9 = 81 ). However, another number can also satisfy this condition: ( -9 ). This is because ( (-9) \times (-9) = 81 ) as well. Thus, the number 81 has two square roots: 9 and -9.
Why are there two square roots?
The concept of a square root involves both positive and negative numbers. This happens because the product of two negative numbers is positive. Therefore, both positive and negative numbers can be multiplied by themselves to yield the same positive result.
In general, for any positive real number ( x ), there are always two square roots: a positive root and a negative root. This can be written mathematically as:
[
\pm \sqrt{x}
]
For 81, the square roots are ( \pm \sqrt{81} = \pm 9 ), which means the two square roots are 9 and -9.
Principal square root:
When dealing with square roots, the principal square root is typically referred to as the positive square root. For example, when you see ( \sqrt{81} ), it is assumed to mean the positive root, 9. However, mathematically, both 9 and -9 are square roots of 81, as they both satisfy the equation:
[
x^2 = 81
]
In summary, every positive number has both a positive and negative square root, and for 81, these are 9 and -9, respectively.