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What are the rules for multiplying algebraic terms?

Published in Algebra Rules 3 mins read

The rules for multiplying algebraic terms involve multiplying the coefficients and applying exponent rules to the variables. Here's a breakdown:

Steps for Multiplying Algebraic Terms:

  1. Separate Coefficients and Variables: Mentally (or physically) separate each term into its numerical coefficient and its variable (pronumeral) part. For example, 3x^2y is separated into 3 and x^2y.

  2. Multiply the Coefficients: Multiply the numerical coefficients of each term together. This gives you the coefficient of the resulting term.

  3. Multiply the Variables: Multiply the variable parts together. This is where the exponent rules come into play:

    • Product of Powers Rule: When multiplying variables with the same base, add their exponents. That is, x^m * x^n = x^(m+n).

  4. Combine the Results: Write the product of the coefficients and the product of the variables together as a single term.

Detailed Explanation with Examples

Here's a more detailed look at the process:

Step Explanation Example
1. Separation Identify the coefficient and the variable part of each term. (4a^2b) * (3ab^3) -> 4 and a^2b, 3 and ab^3
2. Coefficient Multiplication Multiply the numerical coefficients. 4 * 3 = 12
3. Variable Multiplication Apply exponent rules when multiplying the variable parts. a^2b * ab^3 = a^(2+1)b^(1+3) = a^3b^4
4. Combination Combine the multiplied coefficient and variables into a single term. 12a^3b^4

Example 1:

Multiply (2x^3y) * (5xy^2)

  1. Coefficients: 2 and 5
  2. Variables: x^3y and xy^2
  3. Multiply Coefficients: 2 * 5 = 10
  4. Multiply Variables: x^3y * xy^2 = x^(3+1)y^(1+2) = x^4y^3
  5. Combine: 10x^4y^3

Example 2:

Multiply (-3a^2b^4) * (4ab)

  1. Coefficients: -3 and 4
  2. Variables: a^2b^4 and ab
  3. Multiply Coefficients: -3 * 4 = -12
  4. Multiply Variables: a^2b^4 * ab = a^(2+1)b^(4+1) = a^3b^5
  5. Combine: -12a^3b^5

Important Considerations:

  • Order: The order of the terms within the expression doesn't usually matter because multiplication is commutative (a b = b a). However, it's generally good practice to write the coefficient first and then the variables in alphabetical order.
  • Signs: Pay close attention to the signs of the coefficients. A negative times a negative is positive; a negative times a positive is negative.

By following these rules, you can confidently multiply algebraic terms and simplify expressions.

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