Root properties, rational powers and irrational equations
Find the root of a quotient and the root of a root, compare roots of different degrees, and solve an irrational equation by squaring, checking the condition and substituting. Section: Grade 10, nth roots, rational powers, irrational equations; the Grade 8 square root is in another studio.
- Ukraine: grade 10
- Russia: grade 10
Guide
The Ukrainian Grade 10 course (algebra and calculus, §1, sections 5 to 7) teaches the properties of the nth root, powers with rational exponents and irrational equations.
For a ≥ 0, b > 0: ⁿ√(a : b) = ⁿ√a : ⁿ√b, ᵐ√(ⁿ√a) = ᵐⁿ√a, (ⁿ√a)ᵏ = ⁿ√(aᵏ). A factor goes under the root: a · ⁿ√b = ⁿ√(aⁿ · b). To compare roots of different degrees, bring them to a common index.
For a > 0: amn = (n√a)m, and the laws of indices still hold: ap·aq = ap + q, (ap)q = apq.
To solve √f(x) = g(x), square both sides and then check g(x) ≥ 0. Another method is to substitute a new variable.