Understanding Square Roots
In mathematics, the square root of a number $x$ is a number $y$ such that $y^2 = x$. In other words, a square root asks the question: "What number, when multiplied by itself, gives me the original number?"
For example, the square root of $25$ is $5$, because $5 \times 5 = 25$. Finding the square root is the exact inverse operation of taking an exponent of 2. If you need to raise a number to a specific power, you can use our Exponent Calculator instead.
Perfect Squares vs. Imperfect Radicals
Perfect Squares
A perfect square is an integer that is the product of another integer multiplied by itself. When you take the square root of a perfect square, the answer is always a clean, whole number.
$\sqrt{1} = 1$
$\sqrt{4} = 2$
$\sqrt{9} = 3$
$\sqrt{16} = 4$
$\sqrt{100} = 10$
Simplifying Imperfect Radicals
When a number is not a perfect square (like 72), its decimal root goes on infinitely. In algebra, mathematicians prefer to leave these in "simplified radical form" by factoring out the largest perfect square.
Example: Simplify $\sqrt{72}$
1. Find factors: $\sqrt{36 \times 2}$
2. Separate: $\sqrt{36} \times \sqrt{2}$
3. Solve perfect square: $6 \times \sqrt{2}$
Result: $6\sqrt{2}$
Frequently Asked Questions (FAQs)
What is a square root?
The square root of a number $x$ is a value $y$ such that $y$ multiplied by itself gives the original value $x$. For example, the principal square root of $49$ is $7$, because $7 \times 7 = 49$. The mathematical symbol for a square root is the radical sign ($\sqrt{}$).
Can a square root be negative?
Technically, every positive number has two square roots: one positive and one negative. For instance, both $5 \times 5 = 25$ and $-5 \times -5 = 25$. However, the radical symbol ($\sqrt{}$) specifically asks for the principal square root, which is the positive answer. If the negative root is needed, it is written as $-\sqrt{25}$.
Can you take the square root of a negative number?
In the realm of standard real numbers, no, because multiplying any two identical numbers (whether both positive or both negative) always results in a positive product. However, in advanced mathematics, taking the square root of a negative number yields an imaginary number denoted by $i$, where $i = \sqrt{-1}$. Our calculator automatically supports this!