Simplifying Algebraic Expressions: Understanding Squares, Cubes, and Roots

TOPIC 2.3: Algebraic Expressions Section 2.3.2 (c): Simplifying algebraic expressions

Topic 2.3: Algebraic expressions The following sections will be covered: 2.3.1. Algebraic language 2.3.2. Simplifying algebraic expressions 2.3.3. Factorizing algebraic expressions

Simplifying algebraic expressions In Section 2.3.2 learners will be able to: Determine the squares, cubes, square roots and cube roots of single algebraic terms or like algebraic terms. Determine the numerical value of algebraic expressions by substitution.

Squares, cubes and roots Square of a number means multiplying the number by itself twice e.g., the square of 9 = (9)2= 9 9 = 81 Cube of a number means multiplying the number by itself three time e.g., the cube of 4 = 4)3= 4 4 4 = 64 is a symbol for square root. Square root means what positive number multiplied by itself gives the number or term inside the square root. is a symbol for cube root. Cube root means what number when multiplied by itself three times gives the number inside the cube root.

Example: 1 Simplify the following algebraic expressions: (a) (5?3)2 (b) (3?4)3 49?12?2 (c) 364?12 (d) 31 000?9 125?3 (e)

Solutions Example: 1 (a) (5?3)2 = 5?3 5?3 (Remove brackets) = 25?6 the law of exponent for multiplication) (b) (3?4)3 = 3?4 3?4 3?4(Remove brackets) (multiply the numbers i.e. 5 5 = 25 and?3 ?3= ?6 = 27?12(multiply the numbers i.e. 3 3 3 = 27 and ?4 ?4 ?4= ?6 law of exponents for multiplication) the

Solutions Example: 1 (c) 249?12?2 Note: First find the square root of the number i.e. 49 = 7 : Second: find the square root of the variable by dividing the exponent by 2 i.e. ? (d) Note: First find the cube root of the number i.e. : Second: find the cube root of the variable by dividing the exponent by 3 i.e. ? = 4?4 12 2 = ?6and ? 2 2= ?1= y 249?12?2= 7?6y 364?12 364 = 4 12 3 = ?4 364?12

Activity 1 Determine: (a) 203 (b) ( 7?5)3 (c) 5(4?5)2 (d) 121?4?34 (e) 3729?12?9 31000?3+ 144?2 (f)

Solutions: Activity 1 (a) 203= 20 20 20 = 8000 (b) ( 7?5)3= 7?5 7?5 7?5 = -343?15 (c) 5(4?5)2= -5 (4?5 4?5) = -5 16 ?10 = -80 ?10 121?4?34= 11?2?17 (d)

Solutions: Activity 1 (e) 3729?12?9= 9?4?3 31000?3+ = 10x + 12x = 22x 144?2 (f)

Simplifying algebraic expressions The numerical value of algebraic expressions by substitution The numerical value of an algebraic expression depends on the values of the variables. In substituting the numerical values we have to use BODMAS.

Example: 2 (a) Evaluate 3?2 2x if x = -3 (b) Determine the value of the expression ?2+ ?2if a = -2 and b = 3 (c) If x = 3 and y = 4, calculate the value of ?2+ ?2 (d) Evaluate the value of 5(2?2 1) + 10 if p = -5 (e) Determine the value of the expression 15?2 ? +3 if p = -2 2?+1

Solutions: Example 2 (a) 3?2 2x = 3( 3)2 2(-3) = 3(9) + 6 = 27 + 6 = 33 (b) ?2+ ?2= ( 2)2+ (3)2 = 4 + 9 = 13 (c) ?2+ ?2= (3)2+ (4)2 = 9 + 16 = 3 + 4 = 7

Solutions: Example 2 (d) 5(2?2 1) + 10 = 5 2( 5)2 1 + 10 = 5 2 25 1 + 10 = 5 50 1 + 10 = 5(49) + 10 = 245 + 10 = 255 (e) 15?2 ? +3 2?+1 2 2 + 1 = 60 +2 + 3 2( 2)+1 =65 5= 13 = 15( 2)2 2 + 3

Activity: 2 (a) Determine the value of 2(?2+ 1)2 100 if x = -3 (b) Evaluate ?2+ ?2if a = -5 and b = -7 (c) Calculate ?2- m + 6 if m = 1 2

Solutions: Activity 2 (a) 2(?2+ 1)2 100 = 2(( 3)2+ 1)2 100 = 2(9 + 1)2-100 = 2 (10)2- 100 = 2 100 100 = 200 100 = 100 (b) ?2+ ?2= ( 5)2+ ( 7)2 = 25 + 49 = 74

Solutions: Activity 2 (c) ?2- m + 6 = (1 2) -1 = 1 2+ 6 2+ 6 4+ 24 4-1 = 1 1 4-2 = 23 4 4

Final Assessment Questions 1-10 Question 1 100?14 169?6= A: 10?8 13 B: 10?7 14?3 C: 10?7 13?3 D: 1,69?8

Question 2: 2125?6 225?4- A. 5?2 B. 20?2 C. 17?2 D. 10?2

Question 3: Simplify: -5( 4?)2 A. -80?2 A. 16?2 B. 20?2 C. 80?2

Question 4 ( 2? A: 8?3 3)3 27 B: 8?3 27 C: 6?3 27 D: 8?3 9

Question 5 Evaluate 3?2- 2x if x = -3 A: 7 B: -33 C: -21 D: 33

Question 6 121?2?14 A: 11?2?7 B: 11m?12 C: 11m?7 D: 11m?5

Question 7 3216?18?9?3= A: 6?6?3? B: 6?9?3? C: 6?9?6? D: 6?6?3?2

Question 8 Evaluate the algebraic expression -?2- 2? + 2 if a = -2. A: -2 B: 2 C: -6 D: 10

Question 9 -------- of a number means multiplying the number by itself three times. A: square root B: square C: cube D: cube root

Question 10: If s = 3 and t = - 4, determine the value of the expression: 3?2+2? 1 2?2 A. -1 B. 3 17 18 C. D. -2

Solutions: Final Assessment C D A B D C A B C 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. D