Evaluating and Simplifying Expressions

To evaluate an expression with variables, substitute a number for each variable. Then simplify the expression using the order of operations. Be especially careful with exponents and negative signs. For example, the expression − x 2 negative , x squared  always yields a negative or zero value, and ( − x ) 2 open minus . x , close squared  is always positive or zero.

Example 1

Algebra Evaluate each expression for r = 4 . bold italic r equals 4 .

1. − r 2 negative , bold italic r squared

   − r 2 = − 4 2 = − 16 negative , r squared , equals negative , 4 squared , equals , negative 16

2. − 3 r 2 negative 3 , bold italic r squared

   − 3 r 2 = − 3 ( 4 2 ) = − 3 ( 16 ) = − 48 negative 3 , r squared . equals negative . 3 open , 4 squared . close equals negative 3 open 16 close equals , negative 48

3. ( r + 2 ) 2 open bold italic r plus . 2 , close squared

   ( r + 2 ) 2 = ( 4 + 2 ) 2 = ( 6 ) 2 = 36 open r plus . 2 , close squared , equals , open 4 plus . 2 , close squared , equals . open 6 , close squared . equals 36

To simplify an expression, you eliminate any parentheses and combine like terms.

Example 2

Algebra Simplify each expression.

1. 5 r − 2 r + 1 5 bold italic r minus . 2 bold italic r plus 1

   Combine like terms.
   5 r − 2 r + 1 = 3 r + 1 5 r minus . 2 r plus . 1 equals . 3 r plus 1

2. π ( 3 r − 1 ) bold italic pi open 3 bold italic r minus 1 close

   Use the Distributive Property.
   π ( 3 r − 1 ) = 3 π r − π pi open 3 r minus 1 close equals 3 pi r minus pi

3. ( r + π ) ( r − π ) open bold italic r plus , bold italic pi close . open bold italic r minus bold italic pi close

   Multiply polynomials.
   ( r + π ) ( r − π ) = r 2 − π 2 open r plus pi close open r minus pi close equals , r squared , minus , pi squared

Exercises

Algebra Evaluate each expression for x = 5 bold italic x equals 5  and y = − 3 . bold italic y equals . negative 3 .

1. − 2 x 2 negative 2 , x squared
2. − y + x negative y plus x
3. − x y negative x y
4. ( x + 5 y ) ÷ x open x plus 5 y . close , divides x
5. x + 5 y ÷ x x plus . 5 y divides x
6. ( − 2 y ) 2 open minus . 2 y , close squared
7. ( 2 y ) 2 open 2 y , close squared
8. ( x − y ) 2 open x minus . y , close squared
9. x + 1 y fraction x plus 1 , over y end fraction
10. y − ( x − y ) y minus . open x minus y close
11. − y x negative y to the x
12. 2 ( 1 − x ) y − x fraction 2 open 1 minus x close , over y minus x end fraction
13. x · y − x x middle dot . y minus x
14. x − y · x x minus . y middle dot x
15. y 3 − x x − y fraction y cubed , minus x , over x minus y end fraction
16. − y ( x − 3 ) 2 negative y open x minus . 3 , close squared

Algebra Simplify.

17. 6 x − 4 x + 8 − 5 6 x minus . 4 x plus . 8 minus 5
18. 2 ( ℓ + w ) 2 open script l plus w close
19. − ( 4 x + 7 ) negative open 4 x plus 7 close
20. y ( 4 − y ) y open 4 minus y close
21. − 4 x ( x − 2 ) negative 4 x open x minus 2 close
22. 3 x − ( 5 + 2 x ) 3 x minus . open 5 plus 2 x close
23. 2 t 2 + 4 t − 5 t 2 2 , t squared , plus , 4 t minus . 5 , t squared
24. ( r − 1 ) 2 open r minus . 1 , close squared
25. ( 1 − r ) 2 open 1 minus . r , close squared
26. ( y + 1 ) ( y − 3 ) open y plus 1 , close . open y minus 3 close
27. 4 h + 3 h − 4 + 3 4 h plus . 3 h minus . 4 plus 3
28. π r − ( 1 + π r ) pi r minus . open 1 plus pi r close
29. ( x + 4 ) ( 2 x − 1 ) open x plus 4 , close . open 2 x minus 1 close
30. 2 π h ( 1 − r ) 2 2 pi h open 1 minus . r , close squared
31. 3 y 2 − ( y 2 + 3 y ) 3 , y squared , minus , open , y squared . plus 3 y close
32. − ( x + 4 ) 2 negative open x plus . 4 , close squared

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