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Chapter 5: Problem 20

Find each product. $$ -(3 y-8)^{2} $$

Short Answer

Expert verified
-9y^2 + 48y - 64

Step by step solution

01

- Identify the Expression

The given expression is \[-(3y - 8)^2\]
02

- Apply the Square Formula

Use the formula \[ (a - b)^2 = a^2 - 2ab + b^2 \] with \[ a = 3y \] and \[ b = 8 \]
03

- Expand the Expression

Expand the expression using the formula: \[-(3y - 8)^2 = -( (3y)^2 - 2 \times 3y \times 8 + 8^2 ) \]
04

- Simplify Each Term Inside the Parentheses

Calculate each term: \[(3y)^2 = 9y^2 \], \(-2 \times 3y \times 8 = -48y \), and \[8^2 = 64 \]
05

- Combine and Finalize

Combine and simplify the terms inside the parentheses: \[-(9y^2 - 48y + 64)\]
06

- Distribute the Negative Sign

Distribute the negative sign across each term: \[-9y^2 + 48y - 64\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expanding Binomials
When we talk about expanding binomials, we refer to the mathematical process of removing parentheses by multiplying the terms inside. A binomial is an algebraic expression with two terms, typically written in the form (a + b) or (a - b).
This process often involves using special formulas to simplify multiplication.
In our example, we had (3y - 8)^2. Instead of multiplying the binomial by itself directly, we can use a reliable algebraic formula to make our work easier.
Square Formula in Algebra
The square formula in algebra helps simplify binomials raised to a power of 2. It states that: (a - b)^2 = a^2 - 2ab + b^2. This formula lets us quickly expand our expression without manually multiplying each term.
For our specific problem, (3y - 8)^2 , we plugged in a = 3y and b = 8. This substitution allows us to expand (3y - 8)^2 into: 9y^2 - 48y + 64. This streamlined our calculations while avoiding errors.
Simplifying Expressions
Simplifying expressions involves combining like terms to make them more manageable. For our problem, after expanding (3y - 8)^2, we obtained: -(9y^2 - 48y + 64). To simplify, we distributed the negative sign across each term. This means changing the signs of all terms inside the parentheses.
Our expression becomes: -9y^2 + 48y - 64. Each step requires attention to the algebraic rules to ensure accuracy.
Simplifying expressions makes them easier to work with in subsequent calculations, providing a clearer and more concise result.

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Most popular questions from this chapter

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ $$ \text { can be used to perform some multiplication problems. Here are two examples.} $$ $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1^{2} \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 103 \times 97 $$

Match each expression in Column I with the equivalent expression in Column II. Choices in Column II may be used once, more than once, or not at all. An Exercise \(17, x \neq 0 .\). I (a) \(x^{0}\) (b) \(-x^{0}\) (c) \(7 x^{0}\) (d) \((7 x)^{0}\) (e) \(-7 x^{0}\) (f) \((-7 x)^{0}\) II A. 0 B. 1 C. -1 D. 7 E. -7 F. \(\frac{1}{7}\)

Perform each division. $$ \left(2 x^{3}+x+2\right) \div(x+1) $$

Find each product. \(-4 t(t+3)^{3}\)

Simplify by writing each expression with positive exponents. Assume that all variables represent nonzero real numbers. See Example \(5 .\) $$ \frac{\left(m^{7} n\right)^{-2}}{m^{-4} n^{3}} $$
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