nth Roots and Rational Exponents Solve Radical Equations
Objective 1 Electric Field due to a Ring of Charge
Square Roots and Beyond Radicand Radical
Square Roots and Beyond principal square root The positive square root is considered the principal square root
Square Roots and Beyond Index
Square Roots and Beyond Any positive or negative number has one real cube root, and the other two are imaginary
Square Roots and Beyond Index
Exercise 1
Rational Exponents Square, cube, n th roots can be written using rational exponents. In other words, roots have fractional exponents.
Rational Exponents Square, cube, n th roots can be written using rational exponents. In other words, roots have fractional exponents.
Rational Exponents Square, cube, n th roots can be written using rational exponents. In other words, roots have fractional exponents.
Exercise 2 Without a calculator to evaluate the following.
Objective 1 Electric Field due to a Ring of Charge
Real n th Roots In other words, even roots have two solutions, a positive and negative, and the radicands have to be nonnegative.
Real n th Roots Furthermore, odd roots only have one solution, with the same sign as the radicand, which can be positive or negative.
Real n th Roots Odd roots can only have one real solution, all others are imaginary.
More Rational Exponents
Exercise 3 Without a calculator, evaluate the following.
Exercise 4 Use a calculator to approximate the following.
Solving Equations Recall the inverse of squaring a number is taking the square root. Similarly, the inverse of raising a number to the n th power is taking the n th root.
Exercise 5 Solve each equation.
Objective 2 You will be able to solve really cool (radical) equations
Really Cool Equations radical equations The equations below are all examples of radical equations. The radicals involved can be of any index or can even use rational exponents.
Really Cool Equations radical equations The equations below are all examples of radical equations. To solve these awesome equations, you first have to isolate the radical expression, and then raise both sides of the equation to some power to make the radical mathemagically disappear.
Exercise 6 Solve the equation. Check your solution.
Step 3 Step 2 Step 1 Solving Radical Equations To solve radical equations: Isolate the radical Raise each side to some power Solve new polynomial equation Square or cube both sides…
Exercise 7 Solve the equation. Check your solution.
Exercise 8 Solve the equation. Check your solution.
Exercise 9 Solve the equation. Check your solution.
Extraneous Solutions extraneous As the previous exercise demonstrated, it is important to check your solutions because at least one of them may be extraneous. This means that it is an apparent solution that doesn’t actually work in the original equation. Before squaring: After squaring:
Exercise 10 Solve the equation. Check your solution.
Objective 2 You will be able to solve really cool (radical) equations
Step 1 Step 3 Step 2 Step 1 Squaring Ad Nauseum Really radical equations contain more than one radical expression. To solve these equations: Separate radicals Square both sides Isolate radical Solve and check
Exercise 11 Solve the equation. Check your solution.
Exercise 12 Solve the equation. Check your solution.
Rational Exponents Solving equations involving rational exponents is similar to solving radical equations. 1.Isolate the variable/expression with rational exponents 2.Raise both sides to the reciprocal power 3.Solve and check your answer(s)
Exercise 13 Solve the equation. Check your solution.
Exercise 14 Solve the equation. Check your solution.
Substitution
Exercise 15 Solve the equation. Check your solution.
6.1: nth Roots and Rational Exponents 6.6: Solve Radical Equations