Simplify each expression.
35
3 to the fifth
43
4 cubed
24
2 to the fourth
108
10 to the eighth
(23)3
open , 2 thirds , close cubed
(12)4
open , 1 half , close to the fourth
(0.4)6
open 0.4 close to the sixth
74
7 to the fourth
20−2⋅32
20 minus 2 middle dot , 3 squared
6+4÷2+3
6 plus 4 divides 2 plus 3
(62−33)÷2
open , 6 squared , minus , 3 cubed , close divides 2
5⋅22÷2+8
5 middle dot , 2 squared , divides 2 plus 8
80−(4−1)3
80 minus . open 4 minus 1 close cubed
52+82−3(4−2)3
52 plus , 8 squared , minus 3 . open 4 minus 2 close cubed
64÷329
fraction 6 to the fourth , divides , 3 squared , over 9 end fraction
2⋅7+49÷3
fraction 2 dot 7 plus 4 , over 9 divides 3 end fraction
See Problem 3.
Evaluate each expression for s=4
bold italic s equals 4 and t=8.
bold italic t equals 8 .
(s+t)3
open s plus t close cubed
s4+t2+s÷2
s to the fourth , plus , t squared , plus s divides 2
(st)2
open s t close squared
3st2÷(st)+6
3 s , t squared , divides open s t close plus 6
(t−s)5
open t minus s close to the fifth
(2s)2t
open 2 s close squared . t
2st2−s2
2 s , t squared , minus , s squared
2s2−t3÷16
2 , s squared , minus , t cubed , divides 16
(3s)3t+ts
fraction open , 3 s , close cubed . t plus t , over s end fraction
See Problem 4.
Write an expression for the amount of change you will get when you pay for a purchase p with a $20 bill. Make a table to find the amounts of change you will get for purchases of $11.59, $17.50, $19.00, and $20.00.
An object's momentum is defined as the product of its mass m and velocity v. Write an expression for the momentum of an object. Make a table to find the momentums of a vehicle with a mass of 1000 kg moving at a velocity of 15 m/s, 20 m/s, and 25 m/s.
Expression for amount of change:
Change = 20 - p
Table for purchases:
Purchase | Amount of Change
$11.59 | $8.41
$17.50 | $2.50
$19.00 | $1.00
$20.00 | $0.00
Expression for momentum:
Momentum = m * v
Table for momentums:
Velocity (m/s) | Momentum
15 | 15,000 kg·m/s
20 | 20,000 kg·m/s
25 | 25,000 kg·m/s
To simplify each expression, we'll follow the order of operations, which is parentheses, exponents, multiplication/division (from left to right), and addition/subtraction (from left to right).
1. 35 - This expression is already simplified. There are no operations to perform.
2. 3 to the fifth - To simplify this, we need to calculate 3 raised to the power of 5.
Answer: 3^5 = 3 * 3 * 3 * 3 * 3 = 243
3. 43 - This expression is already simplified. There are no operations to perform.
4. 4 cubed - To simplify this, we need to calculate 4 raised to the power of 3.
Answer: 4^3 = 4 * 4 * 4 = 64
5. 24 - This expression is already simplified. There are no operations to perform.
6. 2 to the fourth - To simplify this, we need to calculate 2 raised to the power of 4.
Answer: 2^4 = 2 * 2 * 2 * 2 = 16
7. 108 - This expression is already simplified. There are no operations to perform.
8. 10 to the eighth - To simplify this, we need to calculate 10 raised to the power of 8.
Answer: 10^8 = 10,000,000,000
9. (23)3 - To simplify this, we need to calculate the expression inside the parentheses first.
Answer: 23 = 6, and then 6^3 = 6 * 6 * 6 = 216
10. (2/3)^3 - To simplify this, we need to calculate the expression inside the parentheses first.
Answer: (2/3)^3 = (2/3) * (2/3) * (2/3) = 8/27
11. (12)^4 - To simplify this, we need to calculate the expression inside the parentheses first.
Answer: 12^4 = 12 * 12 * 12 * 12 = 20,736
12. (1/2)^4 - To simplify this, we need to calculate the expression inside the parentheses first.
Answer: (1/2)^4 = (1/2) * (1/2) * (1/2) * (1/2) = 1/16
13. (0.4)^6 - To simplify this, we need to calculate the expression inside the parentheses first.
Answer: (0.4)^6 = 0.4 * 0.4 * 0.4 * 0.4 * 0.4 * 0.4 = 0.001024
14. 74 - This expression is already simplified. There are no operations to perform.
15. 7 to the fourth - To simplify this, we need to calculate 7 raised to the power of 4.
Answer: 7^4 = 7 * 7 * 7 * 7 = 2,401
16. 20 - 2 * 3^2 - To simplify this, we need to perform the exponentiation first, then the multiplication, and finally the subtraction.
Answer: 20 - (2 * 3^2) = 20 - (2 * 9) = 20 - 18 = 2
17. 6 + 4 ÷ 2 + 3 - To simplify this, we need to perform the division first, then the addition.
Answer: 6 + (4 ÷ 2) + 3 = 6 + 2 + 3 = 11
18. (6^2 - 3^3) ÷ 2 - To simplify this, we need to perform the exponentiation first, then the subtraction inside the parentheses, and finally the division.
Answer: (6^2 - 3^3) ÷ 2 = (36 - 27) ÷ 2 = 9 ÷ 2 = 4.5
19. 5 * 2^2 ÷ 2 + 8 - To simplify this, we need to perform the exponentiation first, then the multiplication, then the division, and finally the addition.
Answer: 5 * 2^2 ÷ 2 + 8 = 5 * 4 ÷ 2 + 8 = 20 ÷ 2 + 8 = 10 + 8 = 18
20. 80 - (4 - 1)^3 - To simplify this, we need to perform the subtraction inside the parentheses first, then the exponentiation, and finally the subtraction.
Answer: 80 - (4 - 1)^3 = 80 - (3)^3 = 80 - 27 = 53
21. 5^2 + 8^2 - 3(4 - 2)^3 - To simplify this, we need to perform the exponentiation first, then the subtraction inside the parentheses, then the exponentiation, the multiplication, and finally the addition and subtraction.
Answer: 5^2 + 8^2 - 3(4 - 2)^3 = 25 + 64 - 3(2)^3 = 25 + 64 - 3(8) = 25 + 64 - 24 = 65
22. 64 ÷ 32^9 - To simplify this, we need to perform the exponentiation first, then the division.
Answer: 64 ÷ 32^9 = 64 ÷ (512)^9 = 64 ÷ (512 * 512 * 512 * 512 * 512 * 512 * 512 * 512 * 512) = 64 ÷ 144115188075855872 = 4.44089209850063 * 10^(-16)
23. 2 * 7 + 49 ÷ 3 - To simplify this, we need to perform the multiplication, then the division, and finally the addition.
Answer: 2 * 7 + 49 ÷ 3 = 14 + 49 ÷ 3 = 14 + 16.333333333... = 30.333333333...
Now, let's move on to the next set of problems.
Problem 3: Evaluate each expression for s = 4.
a) (s + t)^3 - To evaluate this expression, substitute s = 4 and the given value of t into the expression.
Answer: (4 + 8)^3 = 12^3 = 1,728
b) s^4 + t^2 + s ÷ 2 - To evaluate this expression, substitute s = 4 and the given value of t into the expression.
Answer: 4^4 + 8^2 + 4 ÷ 2 = 256 + 64 + 2 = 322
c) (s * t)^2 - To evaluate this expression, substitute s = 4 and the given value of t into the expression.
Answer: (4 * 8)^2 = 32^2 = 1,024
d) 3s * t^2 ÷ (s * t) + 6 - To evaluate this expression, substitute s = 4 and the given value of t into the expression.
Answer: 3(4) * 8^2 ÷ (4 * 8) + 6 = 96 + 6 = 102
e) (t - s)^5 - To evaluate this expression, substitute s = 4 and the given value of t into the expression.
Answer: (8 - 4)^5 = 4^5 = 1,024
Problem 4: Write an expression for the momentum of an object.
The expression for the momentum of an object is given by multiplying its mass (m) with its velocity (v). Therefore, the expression for momentum is m * v.
Let's create a table to find the momentums of a vehicle with a mass of 1000 kg moving at different velocities:
Mass (m) | Velocity (v) | Momentum (m * v)
--------------------------------------
1000 kg | 15 m/s | 15000 kg·m/s
1000 kg | 20 m/s | 20000 kg·m/s
1000 kg | 25 m/s | 25000 kg·m/s