Name: _________________________________________________________ Period: ______ Date: ___________________________ 11th Grade Mathematics PSSA Preparation Program o Mastered On: _____________________ Order of Operations and Powers
Anchors Addressed M11.A.3.1.1 – Simplify/evaluate expressions using the order of operations to solve problems (any rational number may be used.) M11.A.2.2.1 – Simplify/evaluate expressions involving positive and negative exponents, roots and/or absolute value (may contain all types of real numbers – exponents should not exceed power of 10.) M11.A.2.2.2 – Simplify/evaluate expressions involving multiplying with exponents (e.g. 𝑥 ! ∙ 𝑥 ! = 𝑥 !" ), powers of powers (e.g. 𝑥 ! ! = 𝑥 !" ), and powers of products 2𝑥 ! ! = 8𝑥 ! (positive exponents only). Concepts
Order of Operations Explanation The order of operations is a set of procedures that, when followed, ensures that all problems are solved the same way to get the same answer. Example 1: Find the value of the expression: 3 ∙ 8 − 4 + 6 Solution: STEP 1: 3 4 + 6 è Simplify (). STEP 2: 12 + 6 è Multiply. STEP 3: 18 è Add. Example 2: Evaluate the expression 2𝑚 + 𝑛, where 𝑚 = 2 and 𝑛 = −3. Solution: STEP 1: Substitute the given values for each variable. 2𝑚 + 𝑛 → 2 2 + (−3) STEP 2: Simplify the expression by using the order of operations. 4 + −3 = 4 − 3 = 1 Calculator Tip: Many calculators available today automatically follow the order of operations. All graphing calculators do as well as many scientific calculators. Before you take a test, know how your calculator handles the order of operations to ensure success.  Multiplication Involving Powers There are two rules involving the multiplication of powers: Multiplication of Powers 𝑥 ! ∙ 𝑥 ! = 𝑥 !!! Raising a Power to a Power 𝑥!
!
= 𝑥 !∙! Example 3: Simplify: 𝑥 ! ∙ 𝑥 ! Solution: When we multiply two variables of the same base, we add the exponents: 𝑥 ! ∙ 𝑥 ! = 𝑥 !!! = 𝑥 ! ! !
Example 4: Simplify: 𝑥 Solution: When raising a power to a power, we multiple the exponents: ! !
𝑥
= 𝑥 !∙! = 𝑥 !" Example 5: Simplify: 2𝑥 ! ! Solution: When raising a power to a power, we multiple the exponents: 2𝑥 ! ! = 2! ∙ 𝑥 !∙! = 8𝑥 !" Division Involving Exponents There are four rules for the division of exponents. When we simplify expressions involving division, we must remove negative exponents by using the negative exponent rule. Dividing Powers Negative Exponents Zero Exponents Power of a Quotient 𝑥!
!!!
=
𝑥
!
𝑥
!!
𝑥 !! =
!
! !!
!
or !!
= 𝑥 ! 𝑥 ! = 1 𝑥
𝑦
!
𝑥!
= !
𝑦
Example 6: Simplify: ! ! Solution: When we divide two variables of the same base, we subtract the exponents: 𝑥!
1
= 𝑥 !!! = 𝑥 !! = ! !
𝑥
𝑥
Since the solution has a negative exponent, flip the fraction and remove the negative sign. Example 7: Simplify: !! !
!!
Solution: When raising a fraction to a power, the power applies to the top and bottom of the fraction: !
!
2𝑥
4𝑥
1 !!! 1 ! 1
=
=
𝑥
= 𝑥 = 4𝑥
16𝑥 ! 4
4
4
Since the exponents are equal to zero when subtracted, the x values are removed. Exercises A. Simplify the following expressions without a calculator. Then, use a calculator to verify your answer. 1. 3 2 − 4 + 12 2. 12 + 6 ÷ 2 + 15 B. Simplify the following expressions. 3. 8 − 12 ! − 16 4. 20 − 10 ! + 15 × 10 5. (!"!!)
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+ 6(2 + 5) 6. 3 24 ÷ 12 − 4 × 4 7. 3 𝑥 + 2 + 4 2𝑥 − 8 9. 3𝑥 + 2 3𝑥 − 3 + 12 11. 4𝑥 ! + 3𝑥 − 10 + 2(𝑥 − 1) 8. 5 2𝑥 ! + 4𝑥 − 6 + 6 − 2𝑥 10. 4𝑥 − 4𝑥 + 4 + 6 12. 2𝑥 ! − 3𝑥 𝑥 + 1 − 12𝑥 C. Evaluate the following expressions if 𝑚 = 4, 𝑛 = −12, 𝑝 = −2, 𝑞 = 7, and 𝑟 = 5. 13. 3 𝑞 − 𝑟 − 𝑚 14. !!!
!
+ 𝑚 15. 𝑞 − 𝑟
16. !"
!
+
!
!
!
+ 𝑞𝑟 − 3 17. 5𝑟 − 2𝑚 + 𝑛𝑝 18. !!!!
!
− 3𝑞 D. Simplify the following expressions. 19. 𝑥 ! ∙ 𝑥 ! 22. −4𝑥 ! ∙ 6𝑥 ! 25. 2𝑥 ! ∙ 4𝑥 ! 20. 𝑥 ! ! 23. 3𝑥 ! ∙ −2𝑥 ! ! 26. −5𝑥 ! 𝑦 𝑥 ! 𝑦 ! 21. 2𝑥 ! ! 24. 6𝑥 ! ∙ −𝑥 ! ! 27. 4𝑥 ! 𝑦 ! ∙ 3𝑥 ! 𝑦 ! E. Simplify the following expressions. !!
28. ! 31. !
29. !!
!
! !"
32. 30. 4𝑥 ! ! !!
! !!
!!! !
!!!
34. !
35. ! 33. −7𝑥 !
!! ! ∙!! !
!!
!! !
! ! ∙! !
!!
36. 𝑥 !! ∙ 𝑥 ! F. Simplify the following expressions. Next to each step, describe the process(es) used. 37. 7! − 4 ÷ 9 + 10!
!" !!
!
Steps Explanation Steps Explanation 38. 3𝑥 ! +
!"!
!! !
+ 𝑥!
!!