Math101 · Independent Ontario learning library
Math101Square Roots
For $a\ge0$, the principal square root $\sqrt a$ is the unique nonnegative number whose square is $a$. Thus $\sqrt{49}=7$, not $\pm7$.
Square roots reverse squaring and express exact geometric lengths. The principal-root convention prevents ambiguity in functions, while the plus-or-minus distinction matters in equations.
Intuition and core definition
For $a\ge0$, the principal square root $\sqrt a$ is the unique nonnegative number whose square is $a$. Thus $\sqrt{49}=7$, not $\pm7$. The equation $x^2=49$ has two solutions, $x=\pm7$; the radical symbol itself names only the principal root.
Notation, language, and conditions
$\sqrt{a}$ has radicand $a$. Over the real numbers, an even root requires a nonnegative radicand. Perfect squares are $0,1,4,9,16,\ldots$. The identity $\sqrt{a^2}=|a|$ for real $a$, not always $a$, because the principal root must be nonnegative.
Why this idea matters
A principal square root names the nonnegative side length whose square is the radicand, distinct from the two solutions that an equation may have.
A dependable method
- Check the domain: for real square roots, require the radicand to be at least zero.
- Look for a perfect square or factor out the largest perfect-square factor.
- Apply the principal-root convention and keep the result nonnegative.
- If solving $x^2=a$, include both signs when $a>0$.
- Square the proposed principal value and compare with the radicand.
