By definition. Wikipedia says that a square root of a number $a$ is a number $y$ such that $y^2$ = $a$. If $y$ is a square root of $a$ according to this definition, then so is $-y$ since $(-y)^2=((-1)y)^2=(-1)^2\cdot y^2=y^2$. It is probably slightly more difficult to explain why there are at most two square roots. The notation $\sqrt{a}$ denotes the principal square root, which by definition is a nonnegative square root.RTCNTC said:Why do we get two answers when taking the square root?
We can, but $0$ and $-0$ is the same number.RTCNTC said:What about 0?
Can we say sqrt{0} = - 0 and 0?
Evgeny.Makarov said:By definition. Wikipedia says that a square root of a number $a$ is a number $y$ such that $y^2$ = $a$. If $y$ is a square root of $a$ according to this definition, then so is $-y$ since $(-y)^2=((-1)y)^2=(-1)^2\cdot y^2=y^2$. It is probably slightly more difficult to explain why there are at most two square roots. The notation $\sqrt{a}$ denotes the principal square root, which by definition is a nonnegative square root.
We can, but $0$ and $-0$ is the same number.
No. A number can have two square roots but the two square roots of a are not "a" and "-a", they are "[tex]\sqrt{a}[/tex]" and [tex]-\sqrt{a}[/tex]" where [tex]\sqrt{a}[/tex] is the squae root I referred to before.RTCNTC said:Why do we get two answers when taking the square root?
For example, let a = any positive number
sqrt{a} = - a and a.
Because [tex](\sqrt{a})^2= a[/tex] by the definition of "square root" and [tex](-\sqrt{x})^2= (-1)^2(\sqrt{a})^2= (1)(a)= a[/tex].Why is this the case?
What about 0?
Can we say sqrt{0} = - 0 and 0?
HallsofIvy said:And, if we allow complex numbers, then the equation [tex]x^3= a[/tex]. where a can be any complex number, has three solutions, [tex]x^4= x[/tex] has four solutions, and, in general, [tex]x^n= a[/tex] has n solutions.
If we restrict ourselves to real numbers, then the equation [tex]x^n= a[/tex], for a any real number and n odd, has one solution, while [tex]x^n= a[/tex], for n even, has 0 solutions if a<0, 1 solution if a= 0, and 2 solutions if a> 0.
A square root is a number that, when multiplied by itself, results in a given number. For example, the square root of 25 is 5, because 5 x 5 = 25. In other words, the square root is the inverse operation of squaring a number.
Yes, both positive and negative numbers can have square roots. However, when calculating the square root of a negative number, the result will be a complex number, which includes an imaginary unit. For example, the square root of -9 is 3i, where i is the imaginary unit.
The positive square root of a number is the principal square root, which is the positive number that, when squared, results in the given number. On the other hand, the negative square root of a number is the negative number that, when squared, also results in the given number. In other words, the positive and negative square roots are two solutions to the same equation.
To simplify a square root, you need to find the largest perfect square factor of the number inside the square root symbol. Then, you can take the square root of that perfect square and move it outside the symbol. For example, the square root of 72 can be simplified to 6√2, because 72 can be broken down into 36 x 2, and the square root of 36 is 6.
The square root and the exponent of 1/2 are essentially the same thing. They both represent the inverse operation of squaring a number. However, the square root is more commonly used when dealing with non-integer exponents, while the exponent of 1/2 is used when dealing with fractional exponents. For example, the square root of 25 is the same as 25^1/2.