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Exponents and Radicals

Exponents and Radicals. Chapter 11. Flashback: One semester ago……. Simplifying Expressions with Integral Exponents. Helpful Tip: Watch the order of operations ….or else! Examples. Sect 11.1 : Simplifying Expressions with Integral Exponents.

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Exponents and Radicals

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  1. Exponents and Radicals Chapter 11

  2. Flashback: One semester ago……..

  3. Simplifying Expressions with Integral Exponents Helpful Tip: Watch the order of operations ….or else! Examples

  4. Sect 11.1 : Simplifying Expressions with Integral Exponents Use one or a combination of the laws of exponents to simplify each expression. State final answers using only positive exponents. Assume all variables represent positive values.

  5. Simplifying Expressions with Integral Exponents

  6. Simplifying Expressions with Integral Exponents

  7. Simplifying Expressions with Integral Exponents

  8. Simplifying Expressions with Integral Exponents

  9. Simplifying Expressions with Integral Exponents

  10. Simplifying Expressions with Integral Exponents

  11. Simplifying Expressions with Integral Exponents

  12. Simplifying Expressions with Integral Exponents

  13. Sect 11.2: Fractional (Rational) Exponents Note: These properties are valid as long as does not involve the even root of a negative number. power root

  14. Fractional (Rational) Exponents When evaluating expressions involving fractional exponents without a calculator:It is usually best to find the root first, as indicated by the denominator, then raise it to the power indicated in the numerator.

  15. Fractional (Rational) Exponents You can evaluate expressions involving fractional exponents using the calculator. Use the key and typeparentheses around the fraction. Example: Evaluateusing the TI-84.

  16. Simplifying Expressions Involving Fractional Exponents The same laws of exponents apply to fractional exponents, though the work may be a little messier.

  17. Simplifying Expressions Involving Fractional Exponents

  18. Simplifying Expressions Involving Fractional Exponents

  19. Simplifying Expressions Involving Fractional Exponents

  20. Sect 11.3: Simplifying Radicals Let a and b represent positive real numbers.

  21. Simplifying Radicals Remember ….

  22. Simplifying Radicals • To reduce a radical to simplest form: • Remove all perfect nth power factors from a radical of order n. • If a fraction appears under the radical or there is a radical in the denominator of the expression, simplify by rationalizing the denominator. • If possible, reduce the order of the radical.

  23. Simplifying Radicals Removing all nth power factors 

  24. Simplifying Radicals Reducing the order of the radical 

  25. Simplifying Radicals Rationalizing the denominator 

  26. Simplifying Radicals Mixed bag…

  27. Sect 11.4: Addition and Subtraction of Radicals To perform addition or subtraction of radicals, you combine like (similar) radicals. Similar radicals differ only in their numerical coefficients (same radicand AND same index).

  28. Addition and Subtraction of Radicals • Plan of Action: • Express each radical in simplest form. • Combine like radicals. Examples

  29. Addition and Subtraction of Radicals

  30. Addition and Subtraction of Radicals

  31. Addition and Subtraction of Radicals

  32. Sect 11.5: Multiplication & Division of Radicals To multiply expressions containing radicals, we will use the property where a and b represent positive values. Notice that the order (indexes) of the radicals being multiplied must be the same.

  33. Multiplication of Radicals Check it out!

  34. Multiplication of Radicals

  35. Division of Radicals We saw earlier that when dealing with an expression containing a radical in the denominator, we had to rationalize the denominator to write it in simplest form. We will now deal with denominators containing two terms.

  36. Division of Radicals To rationalize a denominator that is the sum or difference of two terms, multiply the numerator and denominator of the fraction by the _____________________ of the denominator.

  37. Sect 14.4 Solving Radical Equations To solve radical equations, we will use the fact that That is, we will apply the appropriate inverse operation to “get the variable out of the radical”.

  38. Solving Radical Equations • To solve a radical equation involving one radical: • Isolate the radical expression on one side of the equation. • Raise both sides of the equation to the power that is the same as the order of the radical (inverse operation). • Solve the resulting equation for the variable. • Check for extraneous solutions* by checking the apparent solutions in the original equation. • *Extraneous solutions may be introduced when both sides of an equation are raised to an even power.

  39. Solving Radical Equations

  40. Solving Radical Equations

  41. Solving Radical Equations

  42. Solving Radical Equations WIND POWER The power generated by a windmill is related to the velocity of the wind by the formula where P is the power (in watts) and v is the velocity of the wind (in mph). Find how much power the windmill is generating when the wind is 29 mph.

  43. To solve a radical equation involving two square roots: • Isolate one of the radical expressions on one side of the equation. • Square both sides of the equation. • Simplify. • Isolate the remaining radical expression on one side of the equation. • Square both sides of the equation. • Solve the resulting equation for the variable. • Check for extraneous solutions by checking the apparent solutions in the original equation.

  44. Solving Radical Equations Involving Two Radicals

  45. Solving Radical Equations Involving Two Radicals

  46. End of Section

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