AI Tutor
If a square floor tile has an area of 0.0064Β m20.0064\text{ m}^2, how long is each edge?
Finding that edge means undoing a power, and the operation that reverses an exponent is called a radical.

Parts of a Radical

Every radical expression has two key parts: the radicand (the value trapped under the root symbol) and the index (the small integer indicating which power we are reversing).
When no index is written, it represents an implied square root with an index of 22. The radical symbol x\sqrt{x} denotes only the non-negative principal square root of xβ‰₯0x \ge 0, such that (x)2=x(\sqrt{x})^2 = x.
While the equation x2=49x^2 = 49 has two roots (x=7x = 7 and x=βˆ’7x = -7), the radical expression 49\sqrt{49} uniquely equals 77 because it asks only for the non-negative root.
Odd roots like cube roots are different: since (βˆ’4)Γ—(βˆ’4)Γ—(βˆ’4)=βˆ’64(-4) \times (-4) \times (-4) = -64, we can evaluate βˆ’643=βˆ’4\sqrt[3]{-64} = -4 directly in the real numbers.
How do we handle radicals when the radicand is not a whole number, but a fraction?

Roots of Fractions

By the Quotient Property of radicals, roots of fractions are evaluated by taking the root of the numerator and denominator independently: ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} and ab3=a3b3\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}} for positive integers aa and bb.
A common mistake is trying to split radicals across addition or subtraction like a+b=a+b\sqrt{a + b} = \sqrt{a} + \sqrt{b}, which is false; radicals only distribute cleanly over multiplication and division.