Problem 45 $$\text { Simplify the given alg... [FREE SOLUTION]

Chapter 1: Problem 45

$$\text { Simplify the given algebraic expressions.}$$ $$-4[4 R-2.5(Z-2 R)-1.5(2 R-Z)]$$

Short Answer

Expert verified

The simplified expression is $-24R + 4Z$.

Step by step solution

Distribute the multiplication across the terms

First, note that we need to distribute the negative four across the terms inside the brackets. Before doing so, simplify inside the brackets. Start by expanding the terms, $-2.5(Z-2R)$ and $-1.5(2R-Z)$.

Expand the expressions inside the brackets

Expand $-2.5(Z-2R)$ which gives $-2.5Z + 5R$. Similarly, expand $-1.5(2R-Z)$ which gives $-3R + 1.5Z$. Substitute these into the original expression.

Simplify inside the brackets

Insert back into the expression: $-4[4R - 2.5Z + 5R - 3R + 1.5Z]$. This simplifies inside the brackets to $-4[6R - Z]$.

Apply the outside multiplication

Now distribute $-4$ across the simplified expression within the brackets, $-4(6R - Z)$, which gives $-24R + 4Z$. This is the fully simplified expression.

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Key Concepts

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

Distributive Property

The distributive property is a fundamental concept in algebra that helps in multiplying a single term across terms inside a bracket. It states that for any numbers or expressions, the formula is:

$ a(b + c) = ab + ac $

In our expression, this property allows us to distribute $-4$ across each term within the brackets.
This results in multiplying $-4$ by every term inside the brackets.
It's a useful property that simplifies complex expressions by breaking them into smaller, manageable parts.
By understanding this property, you can work more efficiently with various algebraic problems.

Expanding Expressions

Expanding expressions involves multiplying terms together and eliminating parentheses.
Our step-by-step approach included expanding $-2.5(Z-2R)$ and $-1.5(2R-Z)$.
Let's break these down:

For $-2.5(Z-2R)$, multiply $-2.5$ by both $Z$ and $-2R$ to get $-2.5Z + 5R$.
Similarly, for $-1.5(2R-Z)$, multiply $-1.5$ with each term to obtain $-3R + 1.5Z$.

These expansions simplify the expression within the brackets and make the overall simplification easier to achieve.
Remember that careful attention to signs is crucial when expanding expressions.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining equivalence.
After expanding and distributing, the next step is to combine like terms. This helps in simplifying the expression further.
In our example, once the expression inside the brackets is expanded, we end up with:

$-4[6R - Z]$

Then, distribute the $-4$ across $6R$ and $-Z$.

The results are $-24R$ and $4Z$.

Thus, the simplified expression is $-24R + 4Z$.
Simplifying expressions helps in revealing the essence and form of an algebraic problem. By doing so, problems become easier to understand and solve.

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91影视

$$\text { Simplify the given algebraic expressions.}$$ $$-4[4 R-2.5(Z-2 R)-1.5(2 R-Z)]$$

Short Answer

Step by step solution

Distribute the multiplication across the terms

Expand the expressions inside the brackets

Simplify inside the brackets

Apply the outside multiplication

Key Concepts

Distributive Property

Expanding Expressions

Simplifying Expressions

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