EVALUATING EXPRESSIONS WITH EXPONENTS

Evaluate the following :

Example 1 :

a + b²(b-a) if a = 5 and b = -3

Solution :

Given :

= a + b²(b-a)

By applying the values of a and b, we get

= 5 + (-3)²(-3-5)

= 5 + 9(-8)

= 5 - 72

= -67

So, the answers is -67.

Example 2 :

(x⁴-3wy)/(y³+2w) if w= 4, x = -3 and y = -5.

Solution :

Given :

= (x⁴-3wy)/(y³+2w)

By applying the values of w, x and y, we get

w = 4, x = -3 and y = -5

= (4⁴-3(4)(-5)) / ((-5)³+2(4))

= (4⁴+60) / (-125+8)

= (256+60) / (-117)

= -316/117

So, the answer is -316/117.

Example 3 :

h²k + h(h-k) if h = 4, k = 1/2

Solution :

= 4²(1/2) + 4(4-(1/2))

= (16/2) + 4(7/2)

= 8 + 14

=  22

So, the answer is 22.

Example 4 :

A player's attack percentage A is calculated using the formula 

A = (k - e)/t

where k represents the number of kills, e represents the number of heart attack errors including blocks, and t represents the total attacks attempted. Find the attack percentage when k = 22, e = 11and t = 35.

Solution :

A = (k - e)/t

when k = 22, e = 11and t = 35

A = (22-11)/35

A = 11/35

A = 0.314

So, the players attack percentage is 0.314%.

Example 5 :

Evaluate

5(d+a)/2ab²

when a = -4, b = 1/2, and d = 1/5

Solution :

Given :

= 5(d+a)/2ab²

By applying the given values, we get

= 5(1/5 + (-4)) / 2(-4) (1/2)²

= 5(-19/4) / (-2)

= 95/8

Example 6 :

The formula for the volume V of a cone with radius r and height

h = 1/3 πr²h

Write the expressions for the volume of the cone given below. 

Solution :

From the picture given above,

diameter = 6x, radius = 3x and height = 2x

V = (1/3) πr²h

V = (1/3)π(3x)²(2x)

V = (1/3)π(9x)(2x)

V = 6x²π

Example 7 :

The average price for a movie ticket can be represented by 

P = y²/400 + 7y/100 + 2.96

where y is the number of years since 1980.

(a) Find the average price of a ticket in 1990.

Solution :

P = y²/400 + 7y/100 + 2.96

y = 1990

P = (1990)²/400 + 7(1990)/100 + 2.96

P = 9900.25 + 139.3 + 2.96

P = 2132.51

Example 8 :

Playing time is added at the end of a soccer game to make up for stoppages. An expression for the length of a 90-minute soccer game with x minutes of stoppage time is 90 + x. How long is a game with 4 minutes of stoppage time?

Solution :

Given expression is 90 + x

90 represents number of minutes for soccer game, and x is the minutes for stoppage.

Time taken for the game with 4 minutes

= 90 + 4

= 94 minutes.

Example 9 :

In sphering, a person is secured inside a small, hollow sphere that is surrounded by a larger sphere. The space between the spheres is inflated with air. What is the volume of the inflated space? (The volume V of a sphere is V = 4/3 𝛑 r³. Use 3.14 for 𝛑.)

Solution :

V = 4/3 𝛑 r³

Before inflating :

When r = 2/2 ==> 1

V = 4/3 𝛑 (1)³

= (4/3) (3.14)

= 4.18 m³

After inflating :

When r = 3/2 ==> 1.5

V = 4/3 𝛑 (1.5)³

= (4/3) x 3.14 x 3.375

= 14.13 m³

The space between two spheres = 14.13 - 4.18

= 9.95 m³

Example 10 :

Describe and correct the error in evaluating the expression

Solution :

63 measn 6 should be multiplied three times. Instead of doing that, they have multiplied 6 and 3. That is the error.

Example 11 :

You have a part-time job. One day your boss offers to pay you either 2 ^h − 1 or 2^{h − 1} dollars for each hour h you work that day. Copy and complete the table. Which option should you choose? Explain.

Solution :

When h = 1

When h = 2

When h = 3

When h = 4

When h = 5