Exponential equation

3^x+2 – 3^x+1 + 3^x = 21

3^x 3² -  3^x  3¹  +  3^x = 21

3^x (9-3+1) = 21

3^x × 7 = 21

3^x  = 21/7

3^x =  3¹

x = 1

1.          (a-b)⁰ + 0.1^-2  x  10^-1=

 .          Solution

             1 + ¹/10² x 1/10¹ =

             1 + 1/100 x 1/10 =

             1 + 0.01 x 0.1 =

             1 + 0.001 =

             1.001

             

Expanding logarithms

            Logarithm rules
            1) log_a(mn) = log_a(m) + log_a(n)

2) log_a(^m/_n) = log_a(m) – log_a(n)

3) log_a(mⁿ) = n · log_a(m)

       Example 1

              log₃(5x)=?

          Solution

              log₃(5x) = log₃(5) + log₃(x)

      Example 2

           log₄( ³/_x )=?

         Solution

           log₄( ³/_x ) = log₄(3) – log₄(x)

      Example 3

           log₅(x⁴)=?

         Solution

         log₅(x⁴) = 4 · log₅(x) = 4log₅(x)

      Example 4

            Log₂( ^3a/_7x )=?

         Solution

           Log₂( ^3a/_7x )= log₂(3a ) -  log₂(7x)= 

                              = log₂( 3) + log₂(a ) – log₂(7) – log₂(x )

Solve systems of Linear Equations

1       x - y   = 2

        3x-2y   =9

2       2x  +   3y   =   -4

        5x  +    6y   =   -7

3      x   -  3y   =1

        3x   -9y   =3

system of linear equations

2x + 3y = 19
x + y = 8.

2x+3y=19
x+y=8=>y=8-x
2x+3y=19=>2x+3(8-x)=19
2x+24-3x=19
2x-3x=19-24=-5
x=5
y=8-x=3

Exponents

1.          (a-b)⁰ + 0.1^-2  x  10^-1=

 .          Solution

             1 + ¹/10² x 1/10¹ =

             1 + 1/100 x 1/10 =

             1 + 0.01 x 0.1 =

             1 + 0.001 =

             1.001

             

Factor the polynomials

Exercises

               

         1             x² +7x +12 =

                 x²  +3x +4x +12 =

                 x(x+3) +4(x+3) =

                 (x+3)(x+4)

2.            4x² – 25 =

               (2x -5)(2x + 5)

3.             x³ -64 =

                (x -4)(x² +4x+16)

4            x⁶ – 1=

             (x³ – 1)(x³ + 1) =

             (x – 1)(x² + x +1)((x + 1)(x² –x  +1)

5           25x² – 40xy +16y² =

             (5x – 4y)²

Factoring Polynomials ( Formulas)

Remember:

 A²+2AB + B²=(A+B)²
A²-2AB + B²=(A-B)²

A² - B² = (A + B)(A - B) 

A³+3A² B+3AB²+B³=(A+B)³

A³-3A² B+3AB²-B³=(A-B)³

A³ + B³ = (A + B)(A² - AB + B²) 

Order of operations

 Remember

P E M D  A S

P-        Parentheses

E         Exponents

M       Multiplication

D        Division

A         Addition

S         Subtraction

Example 1

(14-2)×3-12 ÷(6-3)=?

Solution

(14-2)×3-12 ÷(6-3)=

12×3-12÷3=

36-4=

32

Example 2

6²÷3-2²×3+5×3=?

Solution

6²÷3-2²×3+5×3=

36÷3-4×3+5×3=

12-12+15=

0+15=

15

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