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

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

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".

Find the common factor:

\( 2ax+3x \)

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:

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

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

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:**

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

Test your knowledge

Question 1

Find the common factor:

\( ab+bc \)

Question 2

Find the common factor:

\( 7a+14b \)

Question 3

Find the biggest common factor:

\( 12x+16y \)

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’ll explain each of these topics in detail.

**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.**

**Let's return to the essential points within the topic of exponents:**

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:**

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

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Do you know what the answer is?

Question 1

Decompose the following expression into its factors:

\( 26a+65bc \)

Question 2

Decompose the following expression into factors:

\( 37a+6b \)

Question 3

Decompose the following expression into factors:

\( 20ab-4ac \)

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 let’s look at some examples with numbers to understand the formula.

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

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.

Check your understanding

Question 1

Decompose the following expression into factors:

\( 36mn-60m \)

Question 2

Decompose the following expression into factors:

\( 13abcd+26ab \)

Question 3

Which of the expressions is equivalent to the expression?

\( 16-4c \)

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

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.

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?

Question 1

Find the common factor:

\( 2ax+4x^2 \)

Question 2

Find the common factor:

\( 25y-100xy^2 \)

Question 3

Which of the expressions is equivalent to the expression?

\( 8x^2+4x \)

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.

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:**

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

Test your knowledge

Question 1

Choose the expression that is equivalent to the following:

\( 15z^2+50zx \)

Question 2

Which expression is equivalent in value to the following:

\( 99ab^2+81b \)

Question 3

Find the common factor:

\( 2ax+3x \)

Do you know what the answer is?

Question 1

Find the common factor:

\( ab+bc \)

Question 2

Find the common factor:

\( 7a+14b \)

Question 3

Find the biggest common factor:

\( 12x+16y \)

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

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

Decompose the following expression into factors by extracting the common factor:

$xyz+yzt+ztw+wtr$

** Factor** the given expression:

$xyz+yzt+ztw+wtr$We will do this **by extracting the maximum common factor, both from the numbers and the letters,**

We refer to the numbers and letters __separately__, remembering that a common factor is a factor (multiplier) **common to all terms of the expression**,

__As the given expression does not have numeric coefficients (other than 1) we will look for the maximum common factor of the letters:__

There are four terms in the expression:

$xyz,\hspace{4pt}yzt,\hspace{4pt}ztw,\hspace{4pt}wtr$We will notice that in each of the four members there are three different letters, but there is not one or more letters that are included (in the multiplication) in all the terms, that is, there is no common factor for the four terms and therefore it is not possible to factor this expression by extracting a common factor

__Therefore, the correct answer is option d.__

It is not possible to decompose the given expression into factors by extracting the common factor.

Check your understanding

Question 1

Decompose the following expression into its factors:

\( 26a+65bc \)

Question 2

Decompose the following expression into factors:

\( 37a+6b \)

Question 3

Decompose the following expression into factors:

\( 20ab-4ac \)

Related Subjects

- Linear Function
- Graphical Representation of a Function that Represents Direct Proportionality
- The Linear Function y=mx+b
- Finding a Linear Equation
- Positive and Negativity of a Linear Function
- Representation of Phenomena Using Linear Functions
- Inequalities
- Inequalities with Absolute Value
- The Distributive Property
- The Distributive Property for Seventh Graders
- The Distributive Property in the Case of Multiplication
- The Extended Distributive Property
- Algebraic Method
- Absolute Value
- Absolute Value Inequalities