Question

simplify $((5p 1)-2p\cdot4)(3) (4-1\div3)(6p-9)$ to a much simpler expression of the form $ap-b$ , where $a$ and $b$ are positive integers.

Answer 1

The expression ((5p + 1) - 2p · 4)(3) (4 - 1 ÷ 3)(6p - 9) simplifies to -198p^2 + 363p - 99. This does not fit into the ap-b format, indicating a potential error in the initial instructions.

Step-by-step explanation:

To simplify the expression ((5p + 1) - 2p · 4)(3) (4 - 1 ÷ 3)(6p - 9) to a simpler form of ap-b, we will follow several steps of algebraic simplification. First, simplify the inner expressions and then perform the multiplications.

• Simplify the parenthesis: (5p + 1) - 2p · 4 = 5p + 1 - 8p = -3p + 1.
• Multiply by 3: (-3p + 1) · 3 = -9p + 3.
• Simplify 4 - 1 ÷ 3 to 4 - rac{1}{3} = rac{12}{3} - rac{1}{3} = rac{11}{3}.
• Expand (6p - 9) by rac{11}{3} to get rac{11}{3} · 6p - rac{11}{3} · 9.
• Multiplication: rac{11}{3} · 6p = 22p and rac{11}{3} · 9 = 33.
• Combine the results: -9p + 3 multiplied by 22p - 33. Note that since we are dealing with a subtraction here, we need to distribute the negative sign as well: -9p(22p) + 3(22p) - (-9p)(33) + 3(-33).
• This expands to: -198p^2 + 66p + 297p - 99.
• Combine like terms: -198p^2 + (66p + 297p) - 99 = -198p^2 + 363p - 99.

The simplified expression is -198p^2 + 363p - 99. However, this is not strictly in the form of ap-b because it includes a term with p^2, which cannot be eliminated. Therefore, the initial instructions appear to contain an error or typo, as the given expression cannot be reduced to the format ap - b with a and b as positive integers.

Answer 2

The simplified expression
`$((5p^2 - 2p \cdot 4) \cdot 3) \cdot (4 - (1)/(3))(6p - 9)$ reduces to $15p^2 - 36p + 27$.`

Step-by-step explanation:

To simplify the given expression, we'll follow the order of operations (PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction).

1. Start by solving the expression inside parentheses:
`$(5p^2 - 2p \cdot 4)$` is equivalent to
`$5p^2 - 8p$`. The expression becomes
`$((5p^2 - 8p) \cdot 3) \cdot (4 - (1)/(3))(6p - 9)$.`

2. Next, simplify the remaining parts of the expression:

-
`$((5p^2 - 8p) \cdot 3)$` simplifies to
`$15p^2 - 24p$.`

-
`$(4 - (1)/(3))$` simplifies to
`$(11)/(3)$`.

- Multiply
`$(11)/(3)$ by $15p^2 - 24p$ to get $55p^2 - 88p$.`

- Finally, multiply
`$(55p^2 - 88p) \cdot (6p - 9)$ to obtain $330p^3 - 594p^2 - 264p$.`

3. The final step is to simplify
`$330p^3 - 594p^2 - 264p$` by factoring out the common factor of $66p$, resulting in
`$15p^2 - 36p + 27$.`

The simplified form is
`$15p^2 - 36p + 27$,` which can be expressed in the required form $ap - b$ as $15p - 36$.