# Algebraic Method

๐Practice algebraic technique

It is a general term for various tools and techniques that will help us solve more complex exercises in the future.

## Exponents and Powers

Powers are a shorthand way of writing the multiplication of a number by itself several times.

For example:

$4^5=4\times4\times4\times4\times4$

$4$ is the number that is multiplied by itself. It is called the "Base of power".
$5$ represents the number of times the base is multiplied by itself and it is called the "Exponent".

## Test yourself on algebraic technique!

Which of the expressions is equivalent to the expression?

$$16-4c$$

## Distributive Property

This property helps us to clear parentheses and assists us with more complex calculations. Let's remember how it works. Generally, we will write it like this:

$Z\times(X+Y)=ZX+ZY$

$Z\times(X-Y)=ZX-ZY$

## Factoring: This involves taking out the common factor from within the parentheses.

The factoring method is very important. It will help us move from an expression with several terms to one that includes only one.
For example:
$2A + 4B$

This expression consists of two terms. We can factor it by reducin by the greatest common factor. In this case, it's the $2$.
We will write it as follows:

$2A+4B=2\times(A+2B)$

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## Extended Distributive Property

The extended distributive property is very similar to the distributive property, but it allows us to solve exercises with expressions in parentheses that are multiplied by other expressions in parentheses.
It looks like this:

$(a+b)\times(c+d)=ac+ad+bc+bd$

In this article, we will discuss important topics within algebraic methodology. Each of these topics will be explained in more detail in their respective articles.

## Reiteration: Powers

In fact, exponents are a shorthand way of writing the multiplication of a number by itself several times. It looks like this:
$4^5$

$4$ is the number that is multiplied by itself. It is called the Base of the exponent.
$5$ represents the number of times the multiplication of the base is repeated and it is called the Exponent.

That is, in our example:
$4^5=4\times4\times4\times4\times4$

Let's remember that any number raised to the power of $1$ equals the number itself
That is:

$4^1=4$

And remember that any number raised to the power of $0$ equals $1$
$4^0=1$

Mathematical definition to the power of $0$.

An important point to note is the difference between an exponent inside brackets and an exponent outside brackets. For example, what is the difference between

$(-4)^2$ and $-4^2$
It is an important case that could be confusing. When the exponent is outside of the brackets, as in the first case, you have to raise the entire expression to the given exponent, that is

$(-4)^2=(-4)\times(-4)=16$

Conversely, in the second case, one must first calculate the exponent and then deal with the negative sign. That is:

$-4^2=-(4\times4)=-16$

Also remember that exponents come before four of the operations in the order of mathematical operations, but not before parentheses.

For example:
$3\times(4-2)^2=3\times(2)^2=3\times4=12$

`

Do you know what the answer is?

## The Distributive Property

We usually encounter the distributive property around the age of $12$. This property is useful for clearing parentheses and assists with more complex calculations. Let's remember how it works. Generally, we write it as:

$Z \times (X + Y) = ZX + ZY$

$Z \times (X - Y) = ZX - ZY$

Now letโs look at some examples with numbers to understand the formula.

### Example 1: Distributive Property

$6\times26=6\times(20+6)=6\times20+6\times6=120+36=156$

We used the distributive property to solve a problem that would have been more difficult to compute directly.
We can also use the distributive property with division operations.

### Example 2: Distributive Property

$104:4=(100+4):4= 100:4 + 4:4 = 25+1 = 26$

Once again, the distributive property has helped us to simplify a problem that, if solved step by step in a straightforward manner, would have been slightly more complex.

### Example 3: Distributive Property with Variables

Clear the parentheses by applying the distributive property.
$3a\times(2b+5)=$

We will pay close attention to multiplying the term outside the parentheses by each of the terms inside the parentheses according to the correct order of operations.

Do you think you will be able to solve it?

## Factoring: Taking Out the Common Factor from Parentheses

The method of eliminating a common factor is very important. It will help us move from an expression with several terms to one that includes only one.
For example, let's look at the expression:

$2A + 4B$

This expression is now composed of two terms. We can factorize it by eliminating the greatest common term. In this case, it's the number $2$.
We will write it as follows:

$2A+4B=2\times(A+2B)$

We will realize that we have moved from a situation in which we had two parts being added together, to a situation with multiplication. This procedure is called factorization.
We can use the distributive property we mentioned earlier to do the reverse process. Multiply the $2$ by each of the terms inside the parentheses:

In certain cases we might prefer an expression with multiplication, and in other cases one with addition.
In the article that elaborates on this topic, you can see more examples regarding this.

## Extended Distributive Property

The extended distributive property is very similar to the distributive property, but it allows us to solve exercises with expressions in parentheses that are multiplied by other expressions in parentheses.
It looks like this:

$(a+b)\times(c+d)=ac+ad+bc+bd$

How does the extended distributive property work?

• Step 1: Multiply the first term of the first parentheses by each of the terms in the second parentheses.
• Step 2: Multiply the second term of the first parentheses by each of the terms in the second parentheses.
• Step 3: Combine like terms.

Example:

$(a+2)\times(3+a)=$

### Phase 2: Let's multiply the 2 by each of the terms in the second parentheses.

Do you know what the answer is?

### Phase 3: Let's organize the terms and, if there are similar ones, let's associate them.

$(a+2)\times(3+a)=3a+a^2+6+2a=a^2+5a+6$

In the full article about the extended distributive property, you can find detailed explanations and many more examples.

## Examples and exercises with solutions for the Algebraic Method

### Exercise #1

$(a+4)(c+3)=$

### Step-by-Step Solution

When we encounter a multiplication exercise of this type, we know that we must use the distributive property.

Step 1: Multiply the first factor of the first parentheses by each of the factors of the second parentheses.

Step 2: Multiply the second factor of the first parentheses by each of the factors of the second parentheses.

Step 3: Group like terms.

a * (c+3) =

a*c + a*3

4  * (c+3) =

4*c + 4*3

ac+3a+4c+12

There are no like terms to simplify here, so this is the solution!

$ac+3a+4c+12$

### Exercise #2

It is possible to use the distributive property to simplify the expression?

If so, what is its simplest form?

$(a+c)(4+c)$

### Step-by-Step Solution

We simplify the given expression by opening the parentheses using the extended distributive property:

$(\textcolor{red}{x}+\textcolor{blue}{y})(t+d)=\textcolor{red}{x}t+\textcolor{red}{x}d+\textcolor{blue}{y}t+\textcolor{blue}{y}d$Keep in mind that in the distributive property formula mentioned above, we assume that the operation between the terms inside the parentheses is an addition operation, therefore, of course, we will not forget that the sign of the term's coefficient is ery important.

We will also apply the rules of multiplication of signs, so we can present any expression within parentheses that's opened with the distributive property as an expression with addition between all the terms.

In this expression we only have addition signs in parentheses, therefore we go directly to opening the parentheses,

We start by opening the parentheses:

$(\textcolor{red}{x}+\textcolor{blue}{c})(4+c)\\ \textcolor{red}{x}\cdot 4+\textcolor{red}{x}\cdot c+\textcolor{blue}{c}\cdot 4+\textcolor{blue}{c} \cdot c\\ 4x+xc+4c+c^2$To simplify this expression, we use the power law for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

In the next step like terms come into play.

We define like terms as terms in which the variables (in this case, x and c) have identical powers (in the absence of one of the variables from the expression, we will refer to its power as zero power, this is because raising any number to the power of zero results in 1).

We will also use the substitution property, and we will order the expression from the highest to the lowest power from left to right (we will refer to the regular integer as the power of zero),

Keep in mind that in this new expression there are four different terms, this is because there is not even one pair of terms in which the variables (different) have the same power. Also it is already ordered by power, therefore the expression we have is the final and most simplified expression:$\textcolor{purple}{4x}\textcolor{green}{+xc}\textcolor{black}{+4c}\textcolor{orange}{+c^2 }\\ \textcolor{orange}{c^2 }\textcolor{green}{+xc}\textcolor{purple}{+4x}\textcolor{black}{+4c}\\$We highlight the different terms using colors and, as emphasized before, we make sure that the main sign of the term is correct.

We use the substitution property for multiplication to note that the correct answer is option A.

Yes, the meaning is $4x+cx+4c+c^2$

### Exercise #3

It is possible to use the distributive property to simplify the expression below?

What is its simplified form?

$(ab)(c d)$



### Step-by-Step Solution

Let's remember the extended distributive property:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$Note that the operation between the terms inside the parentheses is a multiplication operation:

$(a b)(c d)$Unlike in the extended distributive property previously mentioned, which is addition (or subtraction, which is actually the addition of the term with a minus sign),

Also, we notice that since there is a multiplication among all the terms, both inside the parentheses and between the parentheses, this is a simple multiplication and the parentheses are actually not necessary and can be remoed. We get:

$(a b)(c d)= \\ abcd$Therefore, opening the parentheses in the given expression using the extended distributive property is incorrect and produces an incorrect result.

Therefore, the correct answer is option d.

No, $abcd$.

### Exercise #4

Find the common factor:

$ab+bc$

### Step-by-Step Solution

$ab+bc=a\times b+b\times c$

Let's consider that the common factor is b, so we will remove it:

$b(ab+bc)=$

We divide by b:$b(\frac{ab}{b}+\frac{bc}{b})=$

$b(a+c)$

$b(a+c)$

### Exercise #5

Find the common factor:

$7a+14b$

### Step-by-Step Solution

We divide 14 in a multiplication exercise to help us simplify accordingly:$7\times a+7\times b\times2=$

We extract the common factor 7:

$7(a+2\times b)=7(a+2b)$

$7(a+2b)$