# Distributive Property Calculator + Online Solver With Free Steps

The **Distributive Property Calculator** finds the result of an input expression by using the distributive property (if it holds) to expand it. The generalized distributive property is defined as:

\[ a \cdot (b+c) = a \cdot b+a \cdot c \]

Where $a$, $b$, and $c$ represent some values or even complete expressions. That is, $a$ could be a simple value such as $5$, or an expression $a = 2*pi*ln(3)$.

The calculator supports any number of **variables** in the input. It treats all characters from “a-z” as variables except for ‘i’, which represents mathematical constant iota $i = \sqrt{-1}$. Therefore, you can have $a = pi*r^2$ in the above equation.

## What Is the Distributive Property Calculator?

**The Distributive Property Calculator is an online tool that evaluates the result of an input expression by expanding it via the distributive property, provided that it exists.**

The **calculator interface** consists of a single text box labeled “Expand” in which the user inputs the expression. The input expression may contain values, variables, special operations (logs), mathematical constants, etc.

If the calculator determines the distributive property to hold for the input, it expands the expression using it. Otherwise, the calculator directly solves for the input expression within the parentheses (if any) before applying the outer operator.

## How To Use the Distributive Property Calculator?

You can use the **Distributive Property Calculator **to expand an expression by entering that expression into the text box labeled “Expand.”

For example, suppose we want to evaluate the expression:

\[(5+3x)(3+\ln 2.55) \]

The step-by-step guidelines to do so are:

### Step 1

Enter the input expression into the text box as “(5 + 3x)(3 + ln(2)).” The calculator reads “ln” as the natural log function. Make sure that there are no missing parentheses.

### Step 2

Press the **Submit **button to get the resultant value or expression.

### Results

The result shows up in a new tab and consists of a one-line answer containing the resultant value of the input. For our example, the result tab will have the expression:

\[ 9x + 3x \ln(2) + 15 + 5 \ln(2) \]

#### Variable Inputs

If the input expression contains any variables, the calculator shows the result as a function of those variables.

#### Exact And Approximate Forms

If the input contains defined functions such as the natural logs or square roots, the output will have an additional prompt to switch between the **exact **and **approximate** form of the result.

This option is visible for our example expression. Pressing the approximate form prompt will change the result into a more compact form:

\[ 11.0794x + 18.4657 \]

The approximation is solely due to the floating representation of the result, but up to four decimal places suffice for most problems.

#### When Distributivity Does Not Hold

An example of such a case is $a+(b+c)$ since addition is not distributive and neither is subtraction. Therefore if you input the above expression into the calculator, it will not output a result of the form $(a+b) + (b+c)$. Instead, it will output $a + b + c$.

The above happens because the calculator checks the input for distributivity over the operators before beginning the calculations.

## How Does the Distributive Property Calculator Work?

The calculator works by simply using the definition of distributivity to find the result.

### Definition

The distributive property is a generalization of the distributive law, which states that the following always holds for elementary algebra:

\[ a * (b+c) = a*b + a*c \quad \text{where} \quad a, \, b, \, c \, \in \, \mathbb{S} \]

Where $\mathbb{S}$ represents a set and $*, \, +$ are any two binary operations defined on it. The equation implies that the $*$ (outer) operator **is distributive over** the $+$ (inner) operator. Note that both $*$ and $+$ represent *any* operator, not a specific one.

### Commutativity And Distributivity

Note that the above equation specifically represents the left distributive property. The right distributive property is defined:

\[ (b+c) * a = b*a + c*a \]

Left and right distributivity are different only if the outer operator denoted $*$ is not commutative. An example of an operator that is not commutative is division $\div$ as seen below:

\[ a \div (b+c) = \frac{a}{b} + \frac{a}{c} \neq \frac{b}{a} + \frac{c}{a} \tag*{ (left-distributive) } \]

\[ (b+c) \div a = \frac{b}{a} + \frac{c}{a} \neq \frac{a}{b} + \frac{a}{c} \tag*{ (right-distributive) } \]

Otherwise, as in multiplication $\cdot$, the expressions for left and right distributivity become equal:

\[ a \cdot b + a \cdot c = b \cdot a + c \cdot a \tag*{$\because \, a \cdot b = b \cdot a$} \]

And the property is simply called *distributivity*, implying no distinction between left and right distributivity.

### Intuition

In simple terms, the distributive property states that evaluating the expression within the parentheses before applying the outer operator **is the same as** applying the outer operator to the terms within the parentheses and then applying the inner operator.

Therefore, the order of application of the operators does not matter if the distributive property holds.

### Special Conditions

In the case of **nested brackets**, the calculator expands the expression from the innermost to the outermost. At each level, it checks the validity of the distributive property.

If the distributive property **does not hold **at any nesting level, then the calculator first evaluates the expression within the parentheses in BODMAS order. After this, it applies the outer operator to the result.

## Solved Examples

### Example 1

Given the simple expression $4 \cdot (6+2)$, expand and simplify the result.

### Solution

The given expression involves the distribution of multiplication over addition. This property is valid, so we can expand as follows:

\[ 4 \cdot (6+2) = 4 \cdot 6 + 4 \cdot 2 \]

\[ \Rightarrow 24+8 = 32 \]

Which is the value the calculator shows at the result. We can see that it is equal to the direct expansion:

\[ 4 \cdot (6+2) = 4 \cdot 8 = 32 \]

### Example 2

Consider the following expression:

\[ (3+2) \cdot (1-10+100 \cdot 2) \]

Expand it using the distributive property and simplify.

### Solution

Note that this is a multiplication of two separate expressions $(3+2)$ and $(1-10+100 \cdot 2)$.

In such cases, we separately apply the distributive property for each term in the first expression. Specifically, we take the first term of the first expression and distribute it over the second expression. Then we do the same with the second term and continue until all are exhausted.

If the outer operator is commutative, we can also reverse the order. That is, we can take the first term of the second expression and distribute it over the first and so on.

Finally, we replace each term in the first expression with its distributed result over the second expression (or vice versa in reverse order). Therefore, if we expand the first expression’s terms over the second:

\[ (3+2) \cdot (1-10+100 \cdot 2) = \underbrace{3 \cdot (1-10+100 \cdot 2)}_\text{$1^\text{st}$ term distributed} + \underbrace{ 2 \cdot (1-10+100 \cdot 2)}_\text{$2^\text{nd}$ term distributed} \]

Let us consider the two terms separately for further calculations:

\[ 3 \cdot (1-10+100 \cdot 2) = 3 \cdot 1-3 \cdot 10+3 \cdot 200 = 3-30+600 = 573 \]

\[ 2 \cdot (1-10+100 \cdot 2) = 2 \cdot 1-2 \cdot 10+2 \cdot 200 = 2-20+400 = 382 \]

Replacing these values in the equation:

\[ (3+2) \cdot (1-10+100 \cdot 2) = 573 + 382 = 955 \]

#### Alternate Expansion

Since multiplication is commutative, we would get the same result by expanding the second expression’s terms over the first expression:

\[ (1-10+100 \cdot 2) \cdot (3+2) = [1 \cdot (3+2)]-[10 \cdot (3+2)]+[100 \cdot 2 \cdot (3+2)] \]

### Example 3

Expand the following expression using distributivity and simplify:

\[ \frac{1}{2} \cdot \left [ 5 + \left \{3 + \left (5-7 \right ) \cdot 2 \sqrt{10x} \right \} \right] \]

### Solution

Let $y$ be the input expression. The problem requires the nested application of the distributive property. Let us consider the innermost brackets of $y$:

\[ \left (5-7 \right ) \cdot 2 \sqrt{10x} \]

Applying the right-distributive property of multiplication over addition:

\[ \Rightarrow 5 \cdot 2 \sqrt{10x}-7 \cdot 2 \sqrt{10x} = -4 \sqrt{10x} \]

Substituting this result into the input equation $y$:

\[ y_1 = \frac{1}{2} \left [ 5 + \left \{3-4 \sqrt{10x} \right \} \right] \]

Now we solve for the next pair of brackets in $y = y_1$ :

\[ 5 + \left \{ 3-4 \sqrt{10x} \right \} \]

Since addition is not distributive:

\[ \Rightarrow 5+3-4 \sqrt{10x} = 8-4 \sqrt{10x} \]

Substituting this result into equation $y_1$:

\[ y_2 = \frac{1}{2} \left [ 8-4 \sqrt{10x} \right] \]

Which brings us to the outermost brackets in $y = y_1 = y_2$:

\[ \frac{1}{2} \cdot \left [ 8-4 \sqrt{10x} \right] \]

Applying the left-distributive property of multiplication over addition:

\[ \Rightarrow \frac{1}{2} \cdot 8-\frac{1}{2} \cdot \left (-4\sqrt{10x} \right ) = 4-2 \sqrt{10x} \]

And this is the output of the calculator. Thus:

\[ \frac{1}{2} \cdot \left [ 5 + \left \{3 + \left (5-7 \right ) \cdot 2 \sqrt{10x} \right \} \right] = 4-2 \sqrt{10x} \]

And its approximate form as:

\[ \approx 4-6.32456 \sqrt{x} \]