Expanding Algebraic Skills: Multiplying and Dividing

In algebra, multiplying and dividing terms is about understanding how to work with variables and numbers together. Think of terms like groups of items. Multiplying means you’re adding groups together, and dividing means you’re sharing them evenly. Let's dive into how to multiply and divide algebraic terms

 

Understanding Multiplying and Dividing Terms

Key Parts of Algebraic Terms

• Coefficient: The number part of a term, like the $3$ in $3x$
• Variable: The letter part, like $x$ or $y$
• Power (or Exponent): How many times a variable is multiplied by itself, like $x^2$ (which means $x \times x$.

Like Terms and Different Terms

When multiplying or dividing, you can work with any terms, but it’s easier if they are like terms (terms with the same variable part). Even if they’re not like terms, you can multiply or divide coefficients and variables separately.

Rules for Multiplying Algebraic Terms

1. Multiply the Coefficients: First, multiply the numbers in front.
2. Multiply the Variables: If you’re multiplying the same variable, add their powers.

 

Example 1: Multiplying Like Terms

Problem: Simplify $3x \times 4x$

Solution:

1. Multiply the coefficients: $3 \times 4 = 12$

2. Multiply the variables: $x \times x = x^2$
3. Combine: $$3x \times 4x = 12x^2$$

 

Example 2: Multiplying Different Terms

Problem: Simplify $2x \times 5y$

Solution:

1. Multiply the coefficients: $2 \times 5 = 10$

2. Since $x$ and $y$ are different, keep them as they are.

3. Combine: $$2x \times 5y = 10xy$$

 

 

 

Rules for Dividing Algebraic Terms

1. Divide the Coefficients: Divide the numbers in front.
2. Divide the Variables: If you’re dividing the same variable, subtract the powers.

 

Example 3: Dividing Like Terms

Problem: Simplify $\frac{10x^3}{2x}$

Solution:

1. Divide the coefficients: $\frac{10}{2} = 5$

2. For $x^3 \div x$, subtract the powers: $3 - 1 = 2$, so we get $x^2$
3. Combine: $$\frac{10x^3}{2x} = 5x^2$$

 

Example 4: Dividing Different Terms

Problem: Simplify $\frac{12xy}{4y}$

Solution:

1. Divide the coefficients: $\frac{12}{4} = 3$

2. Cancel out $y$ in both the numerator and denominator, leaving $x$
3. Combine: $$\frac{12xy}{4y} = 3x$$