Radicals and nth roots Solving equations with exponents

Radicals and nth roots Solving equations with exponents and radicals

But first, some Domain and range review. Find the domain and range of the blue graph Find the domain and range of the pink graph

Finding Domain and Range State the domain and range of the function in the red SOLUTION

Find the domain and range of the graph Domain: x ³ 0, Range: y ³ 0 Domain and range: all real numbers

Objectives/Assignment • Change between radical and exponent notation • Evaluate nth roots of real numbers using both radical notation and rational exponent notation. • Use nth roots to solve equations containing radicals and exponents other than 1 or 2.

Ex. 5: Using nth Roots in “Real Life” • The total mass M (in kilograms) of a spacecraft that can be propelled by a magnetic sail is, in theory, given by:

where m is the mass (in kilograms) of the magnetic sail, f is the drag force (in newtons) of the spacecraft, and d is the distance (in astronomical units) to the sun. Find the total mass of a spacecraft that can be sent to Mars using m = 5, 000 kg, f = 4. 52 N, and d = 1. 52 AU.

Solution The spacecraft can have a total mass of about 47, 500 kilograms. (For comparison, the liftoff weight for a space shuttle is usually about 2, 040, 000 kilograms.

Ex. 6: Solving an Equation Using an nth Root • NAUTICAL SCIENCE. The Olympias is a reconstruction of a trireme, a type of Greek galley ship used over 2, 000 years ago. The power P (in kilowatts) needed to propel the Olympias at a desired speed, s (in knots) can be modeled by this equation: P = 0. 0289 s 3 A volunteer crew of the Olympias was able to generate a maximum power of about 10. 5 kilowatts. What was their greatest speed?

SOLUTION • The greatest speed attained by the Olympias was approximately 7 knots (about 8 miles per hour).

Goal 1 Solve equations that contain square roots Solve the equation. Check for extraneous solutions. Simple Radical check your solutions!! Ex. 1) Key Step: To raise each side of the equation to the same power. 6. 6 Solving Radical Equations

One Radical Ex. 3) Don’t forget to check your solutions!! 6. 6 Solving Radical Equations

Radicals with an Extraneous Solution Ex. 5) Don’t forget to check your solutions!! 6. 6 Solving Radical Equations

Two Radicals Ex. 4) Steps for two radical equations • Set radical equal to radical • Square both sides • Solve for x Don’t forget to check your solutions!! 6. 6 Solving Radical Equations

Reflection on the Section Without solving, explain why has no solution. 6. 6 Solving Radical Equations

Radical notation Radical Index Number n>1 Radicand The index number becomes the denominator of the exponent.

More on Radicals • If n is odd a can be any number • If n is even, then a must be a positive number or zero, otherwise there is no real solution.

Example: Radical form to Exponential Form Change to exponential form. or or

Example: Exponential to Radical Form Change to radical form. The denominator of the exponent becomes the index number of the radical.

Using Rational Exponent Notation Rewrite the expression using RADICAL notation. Ex) 6. 1 nth Roots and Rational Exponents

Ex. 2 Evaluating Expressions with Rational Exponents A. OR B. OR Using radical notation Using rational exponent notation.

Evaluate both ways (a) proficient 163/2 and (b) advanced 32– 3/5. SOLUTION Rational Exponent Form a. 163/2= (161/2)3= 43 = 64 b. 1 1 = = 3/5 (321/5)3 32 1 1 = 3 = 8 2 32– 3/5 Radical Form 3 3/2 ( ) 16 = 43 = 64 32 -3/5 1 1 = = 3/5 5 32 ( 32 )3 = 1 8 2

Cancelling radicals to solve equations a. b. c. d.

Equations with “nth root” radicals Ex. 2) Key Step: Before raising each side to the same power, you should isolate the radical expression on one side of the equation. Don’t forget to check it. 6. 6 Solving Radical Equations

3 Goal Ex) 4 Solving Equations by taking the “nth root” of both sides 4 Ex) 5 When the exponent is EVEN you must use the Plus/Minus 5 When the exponent is ODD you don’t use the Plus/Minus 6. 1 nth Roots and Rational Exponents

Solving Equations Ex) 4 4 Take the Square 1 st. Very Important 2 answers ! 6. 1 nth Roots and Rational Exponents

Example: Solve the equation: Note: index number is even, therefore, two answers.

1 x 5 = 512 2 SOLUTION 1 x 5 = 512 2 x 5 = 1024 Multiply each side by 2. x = take 5 th root of each side. 5 x = 4 1024 Simplify.

( x – 2 )3 = – 14 SOLUTION ( x – 2 )3= – 14 (x– 2)= 3 – 14 x = 3 – 14 + 2 x = – 0. 41 Use a calculator.

( x + 5 )4 = 16 SOLUTION ( x + 5 )4 = 16 (x+5) =+ x =+ 4 4 16 16 – 5 take 4 th root of each side. add 5 to each side. x = 2 – 5 or x = – 2 – 5 Write solutions separately. x = – 3 x = – 7 Use a calculator. or

Ex. 4 Solving Equations Using nth Roots A. 2 x 4 = 162 B. (x – 2)3 = 10

Goal 2 Solve equations that contain Rational exponents. Ex. 6) it 6. 6 Solving Radical Equations

Ex: Simplify the Expression. Assume all variables are positive. a. d. b. (16 g 4 h 2)1/2 = 161/2 g 4/2 h 2/2 = 4 g 2 h c.