Subtraction of Powers

Subtracting expressions with powers involves understanding how to handle exponents during the subtraction process. It requires careful attention to the rules of exponents and ensuring that the bases and exponents are treated correctly. There are three components of expressions involving powers:

Coefficients: These are constant values that multiply the base, like 3, -2, etc.

Bases: These are the numbers or variables that are raised to a power, such as x, y, or 2.

Exponents: These are the powers to which the base is raised, such as 2 in x^2 or 3 in 5^3.

When subtracting expressions involving powers, students should follow these rules:

Ensure like bases and exponents: Only subtract terms with the same base and exponent.

Combine coefficients: When bases and exponents are identical, subtract the coefficients.

Simplifying result: Ensure the final expression is simplified by combining all like terms and writing unlike terms as they are.

The following are the methods of subtraction of powers:

Method 1: Horizontal Method

To apply the horizontal method for subtraction of powers, use the following steps:

Step 1: Write both expressions in the same line using a minus sign in between.

Step 2: Ensure bases and exponents are the same to subtract.

Step 3: Subtract the coefficients of like terms. Let’s apply these steps to an example:

Question: Subtract )

Step 1: Write both expressions in the same line,

Step 2: Ensure bases and exponents are the same.

Step 3: Subtract coefficients of like terms: Answer:

Method 2: Column Method

When subtracting expressions using the column method, align terms with the same base and exponent in columns. Then subtract the coefficients for those terms. For example, Subtract )

Solution: Arrange terms with the same base and exponent vertically in columns.

5x² + 3y³ - 2 ← Minuend (from which we subtract) - 2x² - y³ + 4 ← Subtrahend (what we subtract) ----------------------- 3x² + 4y³ - 6

Therefore, upon subtracting, we get 3x^2 + 4y^3 - 6.

In algebra, subtraction involving powers has specific properties:

• Subtraction is not commutative In subtraction, changing the order of the terms changes the result, i.e., A - B ≠ B - A.

• Subtraction is not associative Subtraction cannot be regrouped like addition. When three or more expressions are involved, changing the grouping changes the result. (A − B) − C ≠ A − (B − C)

• Subtraction is the addition of the opposite sign Subtracting an expression is the same as adding its opposite, so to make calculations easier, turn subtraction into addition by changing the signs of the second term. A − B = A + (−B)

• Subtracting zero from an expression leaves the expression as is Subtracting zero from any expression results in the same expression: A - 0 = A.

Tips and tricks are useful for students to efficiently deal with the subtraction of powers. Some helpful tips are listed below:

Tip 1: Always check that the bases and exponents match before subtracting coefficients.

Tip 2: If two terms have identical bases and exponents, subtract their coefficients directly. This simplifies expressions.

Tip 3: Beginners and visual learners can benefit from using the box model or column method to avoid missing signs and mismatching terms.

Subtraction involving powers can be challenging, often leading to common mistakes. However, being aware of these errors can help students avoid them.