Square Roots

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

1. Check the domain: for real square roots, require the radicand to be at least zero.
2. Look for a perfect square or factor out the largest perfect-square factor.
3. Apply the principal-root convention and keep the result nonnegative.
4. If solving $x^2=a$, include both signs when $a>0$.
5. Square the proposed principal value and compare with the radicand.

Worked example

Representations and interpretation

A square of area $a$ has side length $\sqrt a$, making the principal root naturally nonnegative. On a graph, $y=\sqrt x$ begins at the origin and is defined only for $x\ge0$ in the real plane.

Reasoning about variations

The rule $\sqrt{a+b}=\sqrt a+\sqrt b$ is generally false: $\sqrt{9+16}=5$ but $3+4=7$. Products can be split as $\sqrt{ab}=\sqrt a\sqrt b$ when $a,b\ge0$, which is why perfect-square factors may be extracted.

Common mistakes

How to check your work

• Square the simplified nonnegative result and recover the radicand.
• Bound a non-perfect root between consecutive perfect squares.
• Confirm that any extracted factor was a perfect square and that the remaining radicand is simplified.

Practice

1. Simplify $\sqrt{98}$.
2. Evaluate $\sqrt{(-8)^2}$.
3. Solve $x^2=36$.

Answers and brief solutions

Synthesis and transfer

The diagonal of a rectangular screen follows from a squared-length equation; simplifying the radical preserves an exact length while a decimal gives a practical measurement.

For a screen with width $w$ and height $h$, the diagonal is $d=\sqrt{w^2+h^2}$ and is nonnegative because it represents length. If the squared sum contains a perfect-square factor, simplifying first gives an exact radical that can be rounded only at the final measurement stage. Squaring that result checks the arithmetic and returns $w^2+h^2$. This use of the principal root differs from solving $x^2=d^2$, which may yield both $x=d$ and $x=-d$ algebraically. Context discards the negative physical length, but the distinction should be made explicitly rather than hidden inside the radical symbol.

Related topics

• Irrational Numbers
• Radical Expressions
• Pythagorean Theorem

Teaching and accessibility note