Variables and Algebraic Expressions: Extended distributive law

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

• Step 1: Apply the distributive property to expand the expression.

• Step 2: Perform the multiplication for each pair of terms.

• Step 3: Combine any like terms.

Now, let's work through each step:
Step 1: We have the expression $(x + m)(\frac{3}{4} + 5x)$. We'll distribute each term in the first binomial across each term in the second binomial.
Step 2: The expression expands by distributing as follows: $x(\frac{3}{4}) + x(5x) + m(\frac{3}{4}) + m(5x)$.
Step 3: Perform the multiplications: $\frac{3}{4}x + 5x^2 + \frac{3}{4}m + 5mx$.

Note that there are no like terms to combine further.

Therefore, the solution to the problem is $\frac{3}{4}x + 5x^2 + \frac{3}{4}m + 5mx$.

To solve this problem, we'll use the distributive property, which states that for any numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$.

Let's break it down step by step:

Step 1: Apply the distributive property
We will expand the expression $(a+b)(3+\frac{a}{b})$ by distributing the terms in $(a+b)$ over $(3+\frac{a}{b})$.

Step 2: Expand the expression
$(a+b)(3+\frac{a}{b})$ expands as follows:

• First, distribute $a$ to each term in $(3 + \frac{a}{b})$:
• $a \cdot 3 = 3a$
• $a \cdot \frac{a}{b} = \frac{a^2}{b}$
• Next, distribute $b$ to each term in $(3 + \frac{a}{b})$:
• $b \cdot 3 = 3b$
• $b \cdot \frac{a}{b} = a$ (because $b$ cancels with the denominator)

Step 3: Combine and simplify the results
Putting it all together, we have:

$3a + \frac{a^2}{b} + 3b + a$

Simplify the expression by combining like terms:

• $3a + a = 4a$

Thus, the simplified result is:

$4a + \frac{a^2}{b} + 3b$

Therefore, the solution to the problem is $4a + \frac{a^2}{b} + 3b$.

To solve this problem, we'll simplify the expression step by step using the distributive law.

Step 1: Apply the distributive property to the first part of the expression: $(\frac{3}{4} + 2a)(8a + 9ba)$.

• Distribute $\frac{3}{4}$ to $8a$ and $9ba$: $\frac{3}{4} \times 8a = 6a$ and $\frac{3}{4} \times 9ba = \frac{27}{4}ab$.
• Distribute $2a$ to $8a$ and $9ba$: $2a \times 8a = 16a^2$ and $2a \times 9ba = 18a^2b$.

The first part expands to: $6a + \frac{27}{4}ab + 16a^2 + 18a^2b$.

Step 2: Apply the distributive property to the second part of the expression: $(5 + a)(\frac{3}{2}a + b)$.

• Distribute $5$ to $\frac{3}{2}a$ and $b$: $5 \times \frac{3}{2}a = \frac{15}{2}a$ and $5 \times b = 5b$.
• Distribute $a$ to $\frac{3}{2}a$ and $b$: $a \times \frac{3}{2}a = \frac{3}{2}a^2$ and $a \times b = ab$.

The second part expands to: $\frac{15}{2}a + 5b + \frac{3}{2}a^2 + ab$.

Step 3: Simplify the expression by subtracting the second part from the first:

• Combine like terms: $6a - \frac{15}{2}a$ and $\frac{27}{4}ab - ab$.
• Subtract constants and like terms: - $6a - \frac{15}{2}a = -\frac{3}{2}a$. - $\frac{27}{4}ab - ab = \frac{27}{4}ab - \frac{4}{4}ab = \frac{23}{4}ab$. - $(16a^2 - \frac{3}{2}a^2) + (18a^2b - 5b)$.

The full simplified expression is: $-\frac{3}{2}a + \frac{23}{4}ab + \left(16a^2 - \frac{3}{2}a^2\right) + (18a^2b - 5b)$.

Recognize that $16a^2 - \frac{3}{2}a^2 = \frac{32}{2}a^2 - \frac{3}{2}a^2 = \frac{29}{2}a^2$, the final answer is:

The simplified expression is: $-\frac{3}{2}a + 5\frac{3}{4}ab + 14\frac{1}{2}a^2 + (18a^2-5)b$.