Understanding the Distributive Property Step-by-Step
The distributive property is a fundamental concept in algebra that allows us to simplify expressions by multiplying a single term by two or more terms inside a set of parentheses. It's a powerful tool for breaking down complex problems into simpler ones and is essential for solving equations and working with polynomials.
What is the Distributive Property?
In its simplest form, the distributive property states that multiplying a number by a sum (or difference) is the same as multiplying each addend (or subtrahend) by the number and then adding (or subtracting) the products. Mathematically, it is expressed as:
a * (b + c) = a * b + a * c
And similarly for subtraction:
a * (b - c) = a * b - a * c
Here, 'a' is the term being distributed, and 'b' and 'c' are the terms inside the parentheses.
Why is it Important?
Simplifying Expressions: It helps in expanding and simplifying algebraic expressions, making them easier to work with.
Mental Math: It can be used to perform mental calculations more easily. For example, to calculate 7 * 103, you can think of it as 7 * (100 + 3) = 7 * 100 + 7 * 3 = 700 + 21 = 721.
Solving Equations: It's a crucial step in solving many algebraic equations where variables are grouped within parentheses.
How the Calculator Works (Step-by-Step Example)
Let's walk through an example using the calculator above. Suppose we want to evaluate the expression 3 * (4 + 5).
Input Values:
Number to Distribute (a): 3
First Term (b): 4
Second Term (c): 5
When you click "Calculate Steps", the calculator performs the following operations:
Original Expression:
The calculator first displays the expression as entered: 3 * (4 + 5).
Direct Calculation:3 * (9) = 27
Step 1 (Distribute):
The number 'a' (which is 3) is distributed to each term inside the parentheses ('b' and 'c'). This means 3 is multiplied by 4, and 3 is also multiplied by 5. The operation inside the parentheses (addition) is maintained between the new products.
3 * 4 + 3 * 5
Step 2 (Calculate Products):
Next, the individual multiplication operations are performed:
(3 * 4) = 12
(3 * 5) = 15
So the expression becomes: 12 + 15
Step 3 (Calculate Sum):
Finally, the results of the products are added together to get the final simplified value:
12 + 15 = 27
Verification:
The calculator then verifies that the result obtained through distribution (27) matches the result of directly calculating the original expression (27). This confirms the property holds true.
This step-by-step breakdown helps visualize how the distributive property works and reinforces the understanding of this crucial algebraic principle.