Mathematics Exponential Assignment

Task 2:

Question 1a.

Input Equation:
= ((5)^(3))^(2)

Formula used : (a^b)^n= a^(bn)

By expanding exponential form.
= (5^(3))^(2)
= (5^3)^(2)
= (125)^(2)
= 125^(2)
= 125^2
= 15625

Question 1.B

Input Equation:
= ((5^(3))^2)*((5)^(-2))-(5^3)*5
= ((5^3)^2)*((5)^(-2))-(5^3)*5

Formula used : (a^b)^n= a^(bn) and applying BODMAS

By expanding exponential form and applying BODMAS
= ((125)^2)*((5)^(-2))-(5^3)*5
= (125^2)*((5)^(-2))-(5^3)*5
= (15625)*((5)^(-2))-(5^3)*5
= 15625*((5)^(-2))-(5^3)*5
= 15625*(5^(-2))-(5^3)*5
= 15625*(5^-2)-(5^3)*5
= 15625*(0.04)-(5^3)*5
= 15625*0.04-(5^3)*5
= 15625*0.04-(125)*5
= 15625*0.04-125*5
= 625-125*5
= 625-625
=0

Question 1c.

(3-x^-2)/(x^3-x)

Formula used: expanding the denominator and taking x as common and expanding (x^2-1)=(x+1)*(x-1) ; index rule

3 - x(^-2)  =   x(h-2) • (3x^2 - 1) 

X^3 - x  =   x • (x^2 - 1) 

Factoring:  x^2 - 1 

Check : 1 is the square of 1

Check :  x^2  is the square of  x1 Factorization is : (x + 1)  •  (x - 1) 

x(-2) divided by x1 = x((^-2) - 1) = x(^-3) = 1/x3

(3x^2 - 1) /( x^3 • (x + 1) • (x - 1))

Question 2.

1. Move all terms to the left side of the equation
   9x ^ 2 = 81
   Subtract from both sides:
   9x ^ 2 - 81 = 81 - 81
   Simplify the expression
   9x ^ 2 - 81 = 0
   . Factor out the greatest common factor to get perfect squares
   9x ^ 2 - 81 = 0
   Factor out of the terms on the left side:
   9x ^ 2 - 9 * 9 = 0
   9(x ^ 2 - 9) = 0
   X=3,-3

Task 3;

Question 1a.

Apply the logarithm product identity
Log _c( a) + log_c (b )= log_c(ab)
log_15(3) + log_15(5)
log_15(3 * 5)
Simplify the expression
Multiply the numbers
log_15(3 * 5)
log_15(15)
Compute the logarithm of two numbers
=log_15(3 * 5)
=1

Question :1b.

log_2 (18) - 2 log_2(3)

Solve

Apply the logarithm power identity
Log _c(a) -log_c (b) =log_c(a/b)
And log_c (a)+log_c (b)= log_c(ab)
log_2(18)-2 log_2(3)
log_2(18) + log_2(3^-2)

Evaluate the exponent
log_2 (18) + log_2(3^-2)
Apply the logarithm product identity
log _2(18) + log_2(1/9)
log? (18/9)

Compute the logarithm of two numbers
=log_2(18/9)
=1

Question 2a.

In(2x) = 4

Take exponential on both sides

As ln (x) = y And after exponential x = e^y
2x = e^4

Simplify

Divide both sides by the same factor Cancel terms that are in both the numerator and denominator

2x = e^4
X=e^4/2

Question 2b .

e^(2x) = 4

Solve

Take logarithm of both sides

Such as e^y=z
After log : y =ln (z)
e^2x = 4
2x = In(4)

Divide both sides by the same factor
2x = ln(4)
2x /2= In(4)/ 2 =X=ln(4)/2

Cancel terms that are in both the numerator and denominator
2x= In(4)
Apply the logarithm power identity
x = In(4) ^(1/2)
Solution
x = ln(4^(1/2))
X= ln(2)

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