Simplifying Radicals with Variables
Simplify radical expressions with variables in Grade 10 algebra. Use √(xⁿ) = x^(n/2) rules, factor variable expressions, and write simplified radicals with positive exponents.
Key Concepts
When simplifying variables under a radical, all variables represent non negative real numbers. To simplify $\sqrt{x^n}$, the result is $x^{\frac{n}{2}}$.
$\sqrt{x^8} = \sqrt{(x^4)^2} = x^4$ $\sqrt{y^9} = \sqrt{y^8 \cdot y} = \sqrt{y^8}\sqrt{y} = y^4\sqrt{y}$ $\sqrt{20a^5b^2} = \sqrt{4 \cdot 5 \cdot a^4 \cdot a \cdot b^2} = 2a^2b\sqrt{5a}$.
Variables under radicals follow a 'two for one' escape plan for square roots. For every pair of identical variables, one gets to leave the radical house. If there's an odd variable out, it stays inside. This is a fun way to remember you're just dividing the exponent by 2. For example, $x^6$ inside becomes $x^3$ outside.
Common Questions
How do you simplify √(x^n) with variables?
Use the rule √(xⁿ) = x^(n/2). For √(x⁶) = x³. For √(x⁵) = x²√x since 5/2 = 2 remainder 1, giving x² outside and x¹ remaining under the radical.
How do you simplify √(48x³y⁴)?
Factor: √(16 · 3 · x² · x · y⁴) = √16 · √(x²) · √(y⁴) · √(3x) = 4xy²√(3x). Take out pairs from under the radical.
What assumptions apply when simplifying variable radicals?
Variables under radicals are assumed to represent non-negative real numbers, so √(x²) = x (not |x|). This is standard convention in algebra courses.