Expressions and Exponents

1. Expressions

• Consider the following diagram.

This is a square with each side length of \textbf{x} units. We will find the perimeter of this square. We will add all the side lengths to find the perimeter.

Perimeter of the square = x + x + x + x

• Can we simplify this expression further? 

All the terms in the expression are of x terms. We call them like terms.

The algebraic terms with the same variable that are raised to the same power are called like terms. Like terms may have different coefficients.

Let’s consider some sets of like terms.

• Like terms set 1

x, 2x, -17x, \frac{x}{2}

• Like terms set 2

-a, 4a, 100a, \frac{-2a}{3}

Set \mathbf{x} and \mathbf{x^2} are not like terms. Even though they both are of \mathbf{x} variable, they are not raised to the same power.

Now, let’s take the above example here. The perimeter of the square = x + x + x + x. We can simplify further, as they are like terms. We will add the coefficients of like terms keeping the variable unchanged.

The perimeter of the square = x + x + x + x = 4x

• Consider the following diagram. Write an expression for the perimeter of the rectangle and simplify the expression.
  

Let’s construct the expression for the perimeter by adding all the sides.

Perimeter of the rectangle = (x + 1) + (2x - 3) + (x + 1) + (2x - 3)

Now, we can simplify this expression by grouping the x terms and by grouping the constants.

Perimeter of the rectangle = (x + 1) + (2x - 3) + (x + 1) + (2x - 3)

= x + 1 + 2x - 3 + x + 1 + 2x - 3

= x + 2x + x + 2x + 1 - 3 + 1 - 3

= 6x - 4 (Adding x terms and constants respectively)

Therefore, the simplified answer for the perimeter of the rectangle will be \mathbf{6x - 4} units.

• Let’s consider the following expression and simplify it.

  
  (2a - 3a + 1) + (a + b - 4)

Let’s group like terms and constants. After that, when we simplify the expression, as the terms with variable \mathbf{a} and the terms with variable \mathbf{b} and constants are unlike terms, we simplify them separately in a single step.

(2a - 3b + 1) + (a + b - 4)

= 2a + a - 3b + b + 1 - 4

= 3a - 2b - 3

Therefore, \mathbf{3a - 2b - 3} cannot be simplified further and that will be the final answer.

Examples

• Evaluate each of the following

1. (x + 5) + (x - 11)

   = x + 5 + x - 11

   = x + x + 5 - 11

   = 2x - 6

2. (2y - 3) + (y + 5)

   = 2y - 3 + y + 5
   = 2y + y - 3 + 5

   = 3y + 2

3. (4r - 3) + (2r + 1) + (9 - r)

   = 4r - 3 + 2r + 1 + 9 - r

   = 4r + 2r - r - 3 + 1 + 9

   = 5r + 7

4. Jenny is collecting seashells. She collects a certain number of seashells each day for 3 days, and on the fourth day, she collects 5 more than twice the number she collects each day. Write an expression to represent the total number of seashells Jenny collects in 4 days.

Let the number of seashells collected by Jenny in one day be x. Therefore,

The  number of seashells Jenny collects in first 3 days = 3 \times x = 3x

The  number of seashells Jenny collects on the fourth day = 3x + 5

Therefore, the total number of seashells Jenny collects in 4 days = 3x + (3x + 5) 

= 3x + 3x + 5

= 6x + 5

2. Exponents

• Consider the following expression.

a \times a \times a = a^3

We intuitively know the answer is a^3. Here, in a^3, \mathbf{a} is called the base while \mathbf{3} is called the exponent/index/power. Therefore, we can say that base \mathbf{a} is raised to the power \mathbf{3}.

• Consider the following evaluations.

1. a^2 \times a^3 = (a × a) × (a × a × a) = a × a × a × a × a = a^5

2. \frac{a^7}{a^5} = \frac{a× a × a × a × a × a × a}{ a× a} = a × a × a = a^2

3. (a^3)^2 = a^3 \times a^3 = a \times a \times a \times a \times a \times a = a^6

4. \frac{a^2}{a^6} = \frac{a \times a}{a \times a \times a \times a \times a \times a } = \frac{1}{a \times a \times a \times a } = \frac{1}{a^4}

When working out with exponents, there are several rules that will make our evaluations easier. 

1. When expressions with the same base are multiplied, exponents of those expressions are added while keeping the base unchanged.

Example: a^2 \times a^3 = a^{2 + 3} = a^5

2. When expressions with the same base are divided, exponents of the expressions in the denominator are subtracted from the expressions in the numerator while keeping the base unchanged.

Example: \frac{a^7}{a^5} = a^{7 - 5} = a^2

3. When an expression that is already raised to an exponent is further raised to another exponent, two exponents are multiplied together.

Example: (a^3)^2 = a^{3 \times 2} = a^6

4. When an expression with a nonzero base that is raised to a negative exponent, can be expressed as a positive exponent by taking the reciprocal of the base while changing the sign of the exponent.

Example: a^{-3} = \frac{1}{a^3}

Examples

• Simplify each of the following