Square Roots and Radical Rules

What you’ll learn

Simplify square roots with the correct real-number restrictions.

សម្រួលឫសការ៉េដោយគោរពលក្ខខណ្ឌចំនួនពិត។

Meaning and use

The square-root sign denotes the nonnegative principal root. For nonnegative real inputs, a root of a product can be separated. The root of a sum cannot generally be split. The square root of x squared equals the absolute value of x, because the result must be nonnegative.

សញ្ញាឫសការ៉េបង្ហាញឫសមិនអវិជ្ជមាន។ ឫសផលគុណអាចបំបែកក្នុងលក្ខខណ្ឌត្រឹមត្រូវ ប៉ុន្តែមិនអាចបំបែកផលបូកជាទូទៅទេ។

Formulas

Product and quotient

\[\sqrt{ab}=\sqrt a\sqrt b,\qquad \sqrt{\frac ab}=\frac{\sqrt a}{\sqrt b}\]

Root of a square

\[\sqrt{x^2}=\lvert x\rvert\]

Symbols and units

Symbol	Meaning	Unit or restriction
a, b	Real inputs	a ≥ 0 and b ≥ 0; b > 0 in the quotient
x	Any real number in the square rule	Context-dependent
abs(x)	Absolute value of x	Nonnegative magnitude

Conditions and limits

• Keep real-number radicands nonnegative.
• For a quotient, the denominator must not be zero.
• សញ្ញាឫសតែមួយមិនបង្ហាញ ± ទេ; សមីការ x²=a ទើបអាចមានចម្លើយពីរ។

Worked examples

Simplifying a radical

Simplify \(\sqrt{72}\).

1. Factor \(72=36\times2\).
2. Use \(\sqrt{36}\sqrt2=6\sqrt2\).

\(6\sqrt2\)

ដកកត្តាដែលជាការ៉េពេញចេញពីក្រោមឫស។

A negative input before squaring

Evaluate \(\sqrt{(-5)^2}\).

1. Square first: \((-5)^2=25\).
2. Principal root \(\sqrt{25}=5\).

5

ឫសការ៉េគោលតែងមិនអវិជ្ជមាន។

Common mistakes

Instead of	Use	Why
√(9 + 16) = 3 + 4 = 7.	√(9 + 16) = √25 = 5.	A root does not distribute over addition.
√x² = x for every real x.	√x² = abs(x).	For negative x, the absolute value is required.

Practice

Solve these before revealing the worked solutions. Include units where relevant.

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. Simplify √50.
2. Evaluate √((-7)²).
3. Solve x² = 16 over the real numbers.

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. 5√2

   50 = 25 × 2.

2. 7

   The principal square root is nonnegative.

3. x = 4 or x = −4

   Both values square to 16; this differs from the single value √16.

Quick recap

• The principal square root is nonnegative.
• Do not distribute a root over addition.
• Separate root evaluation from solving a squared equation.