Operations on Square Roots

Vocabulary flashcards reviewing fundamental definitions, rules, and equation solutions for operations on square roots.

Square Root of a Positive Number $a$

The positive number $\sqrt{a}$ whose square is equal to the number $a$.

Square Root of a Negative Number

If $a$ is a negative number, then its square root $\sqrt{a}$ does not exist.

Basic Identity of Square Roots

For any positive number $a$, $(\sqrt{a})^2 = a$ and $\sqrt{a^2} = a$.

Product Rule for Square Roots

For any two positive numbers $a$ and $b$, $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.

Quotient Rule for Square Roots

For any two positive numbers $a$ and $b$ where $b \neq 0$, $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

Sum of Square Roots Non-Equivalence

The sum of two square roots is not equal to the square root of their sum; for example, $\sqrt{2} + \sqrt{3} \neq \sqrt{5}$.

Simplifying a Sum of Square Roots

A technique that involves simplifying each radical term into a product containing a perfect square and then factoring out the common radical.

Solutions to $x^2 = a$ for $a > 0$

If $a > 0$, the equation $x^2 = a$ has two distinct solutions: $\sqrt{a}$ and $-\sqrt{a}$.

Solutions to $x^2 = a$ for $a < 0$

If $a < 0$, the equation $x^2 = a$ accepts no real solutions.

Solutions to $x^2 = a$ for $a = 0$

If $a = 0$, the equation $x^2 = a$ accepts exactly one solution: $0$.