Extended Distributive Property: Ascertain whether the law of distributive property is applicable

To solve the problem and apply the distributive property correctly, follow these steps:

• Identify the expression, which is $a(b+c)$.
• Apply the distributive property: multiply $a$ by each term inside the parentheses.

Applying this, we get:

• $a \times b = ab$
• $a \times c = ac$

Combine these two products:

The simplified expression is: $ab + ac$.

This matches with answer choice 2: Yes, the answer $ab+ac$.

To solve this problem, we must determine if we can apply the distributive property to simplify the expression $(a+b)(c \cdot g)$.

The distributive property states that for any three terms, the expression $x(y+z)$ results in $xy + xz$. Here, we have the sum $(a + b)$ and the product $(c \cdot g)$.

We can treat $(c \cdot g)$ as a single term because it involves multiplication, which makes it like a single number or variable in terms of manipulating the expression algebraically. Therefore, using the distributive property, we distribute $(c \cdot g)$ over the terms within the parentheses:

• Step 1: Distribute $c \cdot g$ to $a$, yielding $acg$.
• Step 2: Distribute $c \cdot g$ to $b$, yielding $bcg$.

Hence, the simplified expression is:

$acg + bcg$.

Therefore, the correct answer, according to the choices provided, is:

No, $acg + bcg$.

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.

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.

To solve the problem, we will use the distributive property. Our goal is to expand and simplify the given expression by distributing each term separately:

• Step 1: Multiply the first term of the first binomial, $3a$, by each term in the second binomial $(2x+4)$:
  $3a \cdot 2x = 6ax$
  $3a \cdot 4 = 12a$
• Step 2: Multiply the second term of the first binomial, $-2$, by each term in the second binomial $(2x+4)$:
  $-2 \cdot 2x = -4x$
  $-2 \cdot 4 = -8$
• Step 3: Combine all the products to write the expanded expression:
  $6ax + 12a - 4x - 8$

Therefore, the simplified expression using the distributive property is $6ax + 12a - 4x - 8$.

Thus, the correct answer is Yes, $6ax+12a-4x-8$.

Let's analyze the expression step-by-step:

The original expression is $(x+3)4x + 2$.

• Step 1: Apply the distributive property to the first part of the expression, $(x+3)4x$.
• First, distribute $4x$ to $x$:
  $4x \cdot x = 4x^2$.
• Then, distribute $4x$ to $3$:
  $4x \cdot 3 = 12x$.
• Therefore, by applying the distributive property, $(x+3)4x = 4x^2 + 12x$.
• Step 2: Add the remaining term in the expression, which is $+ 2$.

Combining all the parts together gives:

$4x^2 + 12x + 2$

With these calculations, we can clearly see that the distributive property has been applied correctly and the fully simplified expression is:

$4x^2 + 12x + 2$

Reviewing the multiple-choice answers, the option that aligns with our calculated expression and indicates a "No" response for incorrectly applying distributive property is:

No, $( 4x^2 + 12x + 2 )$

Thus, the correct choice is option 2.

To solve this problem, we'll follow these steps:

• Step 1: Distribute the expressions inside the parentheses.
• Step 2: Multiply and simplify expressions inside first, then outside.

Let's work through each step:

Step 1: Consider the expression $(ab-7)(3+a)$. Apply the distributive property:

• First, distribute $ab$ to both terms inside the second parentheses:
 $ab \cdot 3 = 3ab$
 $ab \cdot a = a^2b$

• Next, distribute $-7$ to both terms inside the second parentheses:
 $-7 \cdot 3 = -21$
 $-7 \cdot a = -7a$

Combining these, we have:

$(ab - 7)(3 + a) = 3ab + a^2b - 21 - 7a$.

Step 2: Multiply through by the factor $2$ outside the parentheses:

Thus, our expression becomes:

$6ab + 2a^2b - 42 - 14a$.

Therefore, the solution to the problem is $6ab + 2a^2b - 42 - 14a$.

To solve the problem, we first examine whether the distributive property can be applied to the expression $(x+y)7+m$.

• The expression $(x+y)7$ involves multiplying the sum of $x$ and $y$ by 7.
• According to the distributive property, we can write this as $7x + 7y$.
• Therefore, the expression simplifies to $7x + 7y + m$.
• Note that the term $m$ is separate and does not involve multiplication; it remains as an addition to the other terms.

Thus, the correct simplified form is $7x + 7y + m$.

Checking the multiple-choice options, the correct choice is:

No, $7x + 7y + m$

Let's simplify the expression $(a+b)\cdot4\cdot(b+2)$ using the distributive property.

Step 1: Distribute the $4$ across $(b + 2)$.
$4(b + 2) = 4b + 4 \times 2 = 4b + 8$

Step 2: Now distribute this result $(4b + 8)$ across $(a + b)$.
$(a+b)(4b+8) = a(4b+8) + b(4b+8)$

Step 3: Apply the distributive property again for both terms.
- For $a(4b+8)$, we get:
$a \times 4b + a \times 8 = 4ab + 8a$
- For $b(4b+8)$, we get:
$b \times 4b + b \times 8 = 4b^2 + 8b$

Step 4: Combine all parts.
The expanded expression is:
$4ab + 8a + 4b^2 + 8b$

Therefore, the simplified expression is $\boxed{4ab + 8a + 4b^2 + 8b}$, and the correct choice is:

Yes, $4ab + 8a + 4b^2 + 8b$.

To simplify the expression $a(b+c)(-b-c)$ using the distributive property, follow these steps:

• Step 1: Apply Distributive Property to $(b+c)(-b-c)$:
  The expression $(b+c)(-b-c)$ can be expanded using the distributive property:
  $(b+c)(-b-c) = b(-b-c) + c(-b-c)$.
• Step 2: Simplify Each Part:
  Let's simplify each term individually:
  - $b(-b) = -b^2$
  - $b(-c) = -bc$
  - $c(-b) = -bc$
  - $c(-c) = -c^2$
  So, combining these results:
  $(b+c)(-b-c) = -b^2 - bc - bc - c^2 = -b^2 - 2bc - c^2$.
• Step 3: Distribute $a$ Over the Result:
  Now, apply a further distribution of $a$ to get:
  $a(-b^2 - 2bc - c^2) = a(-b^2) + a(-2bc) + a(-c^2)$.
• Step 4: Simplify:
  Perform the distribution:
  - $a(-b^2) = -ab^2$
  - $a(-2bc) = -2abc$
  - $a(-c^2) = -ac^2$
  Thus, the expression simplifies to:
  $-ab^2 - 2abc - ac^2$.

Therefore, the simplified expression using the distributive property is $-ab^2 - 2abc - ac^2$.

Given the multiple-choice options, the correct choice that corresponds to our derived expression is:

Choice 4: Yes, $-ab^2 - 2abc - ac^2$

We may use the parenthesis on the right hand side due to the fact that it can be simplified as follows :

(8+a)

Resulting in the following calculation:

$(17+c)(8+a)=$

$136+17a+8c+ca$

We begin by opening the parentheses using the distributive property in order to simplify the expression:

$x(y+z)=xy+xz$Note that in the distributive property formula we assume that there is addition between the terms inside of the parentheses, therefore it is crucial to take into account the sign of the coefficient of the term.

Furthermore, we apply the rules of multiplication of signs in order to present any expression within the parentheses. The parentheses are opened with the help of the distributive property, as an expression in which there is an addition operation between all the terms:

$(3a-4)b+2\\ \big(3a+(-4)\big)b+2$We continue and open the parentheses using the distributive property:

$\big(3a+(-4)\big)b+2\\ 3a\cdot b+(-4)\cdot b +2\\ 3ab-4b+2$Therefore, the correct answer is option c.

To solve this problem using the distributive property, let's expand the given expression $(a+c+d)(a+e)$ step by step.

Step 1: Expand $(a+c+d)(a+e)$ using the distributive property:

• Distribute $a$ over $(a+e)$:
  $a(a+e) = a^2 + ae$
• Distribute $c$ over $(a+e)$:
  $c(a+e) = ca + ce$
• Distribute $d$ over $(a+e)$:
  $d(a+e) = da + de$

Step 2: Combine all distributed terms:

$a^2 + ae + ca + ce + da + de$

Thus, the expression simplifies to $\bm{a^2 + ae + ca + ce + da + de}$.

Therefore, the solution to the problem is Yes, $(a^2 + ae + ca + ce + da + de)$, which matches choice 3.