To evaluate the given expressions with exponents, we have to be clear with the following concepts.
Evaluate the following :
Example 1 :
a + b^{2}(b-a) if a = 5 and b = -3
Solution :
Given :
= a + b^{2}(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 :
(x^{4}-3wy)/(y^{3}+2w) if w= 4, x = -3 and y = -5.
Solution :
Given :
= (x^{4}-3wy)/(y^{3}+2w)
By applying the values of w, x and y, we get
w = 4, x = -3 and y = -5
= (4^{4}-3(4)(-5)) / ((-5)^{3}+2(4))
= (4^{4}+60) / (-125+8)
= (256+60) / (-117)
= -316/117
So, the answer is -316/117.
Example 3 :
h^{2}k + h(h-k) if h = 4, k = 1/2
Solution :
= 4^{2}(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)/2ab^{2 }
when a = -4, b = 1/2, and d = 1/5
Solution :
Given :
= 5(d+a)/2ab^{2}
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 πr^{2}h
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) πr^{2}h
V = (1/3)π(3x)^{2}(2x)
V = (1/3)π(9x)(2x)
V = 6x^{2}π
Example 7 :
The average price for a movie ticket can be represented by
P = y^{2}/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 = y^{2}/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
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