nth Roots and Rational Exponents Solve Radical Equations

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 • 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, nth roots can be written using rational exponents. In other words, roots have fractional exponents.

Rational Exponents Square, cube, nth roots can be written using rational exponents. In other words, roots have fractional exponents.

Rational Exponents Square, cube, nth 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 nth Roots In other words, even roots have two solutions, a positive and negative, and the radicands have to be nonnegative.

Real nth Roots Furthermore, odd roots only have one solution, with the same sign as the radicand, which can be positive or negative.

Real nth 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 nth power is taking the nth root.

Exercise 5 Solve each equation.

Objective 2 You will be able to solve really cool (radical) equations

Really Cool 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 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.

Solving Radical Equations To solve radical equations: Isolate the 1 Step radical Square or cube both sides… Raise each Stepside 2 to some power Solve new polynomial Step 3 equation

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 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

Squaring Ad Nauseum Really radical equations contain more than one radical expression. To solve these equations: Solve and 1 Step check Separate Step 1 radicals Isolate Step 3 radical Square both 2 Step sides

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.

nth Roots and Rational Exponents Solve Radical Equations •