Solving Equations with Exponents and Radicals

Explore the concepts of radicals and nth roots in solving equations involving exponents and radicals. Understand how to find the domain and range of functions graphically. Practice changing between radical and exponent notation, evaluating nth roots of real numbers, and solving real-life problems using nth roots. Dive into examples like determining the total mass of a spacecraft and finding the maximum speed of a ship based on power generated by a crew. Enhance your understanding of mathematical principles through practical applications.

• Exponents
• Radicals
• Equations
• Nth Roots
• Real-life Applications

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1. Radicals and nth roots Solving equations with exponents and radicals

2. But first, some Domain and range review. Find the domain and range of the blue graph ?: < ? < R: 4 ? < Find the domain and range of the pink graph ?: < ? < R: y= 1.2

3. Finding Domain and Range State the domain and range of the function in the red SOLUTION ?: 2 ? < R: < ? 1

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

5. 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.

6. 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:

7. 2 . 0 015 fd m M = 4 3 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.

8. 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.

9. 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 = 0.0289s3 A volunteer crew of the Olympias was able to generate a maximum power of about 10.5 kilowatts. What was their greatest speed?

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

11. Solve equations that contain square roots 1 1 Goal Solve the equation. Check for extraneous solutions. Key Step: Simple Radical = 3 x Ex.1) To raise each side of the equation to the same power. ( ) ( ) 3 2 2 = check your solutions!! x = 9 x 6.6 Solving Radical Equations

12. One Radical + = ( 2 46 ) 8 4 6 + + = 2 8 4 4 6 Ex.3) x + =( )2 = 100 92 4 10 10 = = = 100 4 6 2 + x 2 + x 2 + ( ) 8 8 2 = x 2 = x 8 8 2 10 4 6 x 8 2 Don t forget to check your solutions!! 46 6.6 Solving Radical Equations

13. Radicals with an Extraneous Solution x 3= Ex.5) = 4 x 9 3 ) 9 ( 4 ( ) ( 6 4 x )2 9 2 = = 3 + x 4x x x 4 = = x 2 x ( 4 9 ) 1 2 x x + 10 9 x x 0 = 1 3 ) 1 ( 4 )( 0 = = 9 1 x x Don t forget to check your solutions!! 6.6 Solving Radical Equations

14. Two Radicals Steps for two ) 2 ( 2 radical equations Set radical equal to radical Square both sides Solve for x = 12 2 2 0 + = 12 2 2 2 0 Ex.4) x x + 2 2 x =( = x x 2 x = 8 2 2 0 ) 12 12 ( ) 2 + x 2 x x 2 2 2 x 4 = 2 + 12 2 = 2 2 2 2 0 x x 12 = 6 2 6 x 6 x Don t forget to check your solutions!! = 6.6 Solving Radical Equations

15. Reflection on the Section Without solving, explain why + = 2 4 8 x has no solution. 6.6 Solving Radical Equations

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

17. More on Radicals na 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.

18. Example: Radical form to Exponential Form Change to exponential form. 23 = 2 3x x ( or ) 2 13 = x or ( ) x 1 = 2 3

19. Example: Exponential to Radical Form Change to radical form. ( ) 23 2 x = 2 3 3 or x x The denominator of the exponent becomes the index number of the radical.

20. Using Rational Exponent Notation Rewrite the expression using RADICAL notation. ( )3 24 1 324 Ex) ( )4 28 1 428 Ex) 6.1 nth Roots and Rational Exponents

21. Ex. 2 Evaluating Expressions with Rational Exponents = = = 3 3 3 9 ( 9 ) 3 27 A. Using radical notation 2 Using rational exponent notation. 1 2 2 3 1 = = = 3 3 9 9 ( ) 3 27 1 OR 2 2 1 1 2 B. = = = = 32 5 2 2 4 32 5 ( 32 ) 5 1 1 1 2 = = = 32 OR 5 2 1 2 2 4 32 ( ) 5

22. Evaluate both ways (a) proficient 163/2 and (b) advanced323/5. SOLUTION Radical Form 163/2 ( )3 = Rational Exponent Form a. 163/2 (161/2)3 = b. 32 3/5 =1 64 1 = = 43 = 43 = 64 16 1 1 = = = 32-3/5 (321/5)3 323/5 5 32 323/5 ( )3 =1 1 8 = 23 1 8 =1 = 23

23. Cancelling radicals to solve equations x a. = 2 x x = 6 6 x b. y = 11 y 11 c. = 8 4 r r = r 4 4 4 4 4 4 4 r r d. = r r = 2 r

24. Equations with nth root radicals + = 3 6 6 12 x Key Step: Ex.2) 6 = = Before raising each side to the same power, you should isolate the radical expression on one side of the equation. 3x 3x 6 ( ) ( )3 6 3 = x 216 Don t forget to check it. 6.6 Solving Radical Equations

25. Solving Equations by taking the nth root of both sides 3 3 Goal 4 4 4= 81 x 5= Ex) 2 64 x Ex) 5 5 = 481 x 5= 32 x = 532 x = 3 x = 2 x When the exponent is EVEN you must use the Plus/Minus When the exponent is ODD you don t use the Plus/Minus 6.1 nth Roots and Rational Exponents

26. Solving Equations 4 4 ) 4 4= Take the Square 1st. ( 256 x Ex) = 4256 4 x Very Important 2 answers ! = 4 4 x x = 0 4 = 4 x x 4 = = 4 x 8 6.1 nth Roots and Rational Exponents

27. Example: Solve the equation: x = 7 7 x x 4 7 9993 9993 7 10000 + = + 4 x = = = 4 4 4 410000 10 Note: index number is even, therefore, two answers. x

28. 1 2 x5 = 512 SOLUTION 1 2 x5= 512 x5= 1024 Multiply each side by 2. x = 5 1024 take 5th root of each side. x = 4 Simplify.

29. ( x 2 )3 = 14 SOLUTION ( x 2 )3= 14 ( x 2 ) = 3 14 x = 3 14 + 2 x = 3 14 + 2 x = 0.41 Use a calculator.

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

31. Ex. 4 Solving Equations Using nth Roots A. 2x4 = 162 B. (x 2)3 = 10 2) (x = 10 = 3 = 4 2 162 x 3 x - = 2 10 + = 4 81 x 3 10 2 x = 4 81 x . 4 15 x = 3 x

32. Solve equations that contain Rational exponents. 2 2 Goal 3 2 = 3 2 ( 2 25 ) 250 = Ex. 6) 2 250 x 2 2 = = 125 ( 125 3x 2 ) ( ) 2 3 2 3 3x 2 3) 1 2 ( 2 25 ( 125 ( )2 5 25 ) 2 1) 2 15625 ( 2 = 1 3 x = x ( 2 125 ) = = 250 x it 6.6 Solving Radical Equations

33. Ex: Simplify the Expression. Assume all variables are positive. a. = 3 3z = 9 3 2 3 9 27z 327 z 2 1 18 rs 3 1 d. = 3 3 r s t 3 4 1 t 3 6 r 4 b. (16g4h2)1/2 = 161/2g4/2h2/2 = 4g2h 2 3 = 3 3 r s t 3 4 c. 5 5 5 x x = 5 10 y 10 y 5 x = 2 y