/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 81 Simplify each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 81

Simplify each expression. $$ -\frac{5}{4}\left(-\frac{4}{5} x\right) $$

Short Answer

Expert verified
The simplified expression is \( x \).

Step by step solution

01

Identify the expression

The given expression is \[\begin{equation} -\frac{5}{4}\left(-\frac{4}{5} x\right)\end{equation}\]
02

Distribute the multiplication

Distribute the multiplication of \[\begin{equation}-\frac{5}{4} \text{ and } -\frac{4}{5}x\end{equation}\]
03

Simplify the fractions

Multiply the fractions first: \[\begin{equation}-\frac{5}{4} \cdot -\frac{4}{5} = \frac{5\cdot4}{4\cdot5} = 1\end{equation}\]
04

Calculate the final expression

Now we multiply 1 by x: \[\begin{equation}1 \times x = x\end{equation}\]

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Key Concepts

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

Using the Distributive Property in Algebra
The distributive property is a vital concept in algebra. It allows you to multiply a single term by each term inside a parenthesis. For the given problem, the distributive property helps us simplify the expression \(-\frac{5}{4}\left(-\frac{4}{5} x\right)\). This means we multiply \(-\frac{5}{4}\) by \(-\frac{4}{5}x\). Using the distributive property not only makes complex equations more manageable but also ensures accuracy. This rule can be applied across various scenarios including algebra and arithmetic.
How to Multiply Fractions in Algebra
Fraction multiplication might seem tricky, but it's straightforward with practice. To multiply two fractions, you multiply their numerators together and their denominators together. Let's take the numerators \-5\ and \-4\ from \(-\frac{5}{4}\) and \(-\frac{4}{5}x\) respectively. Multiplying these gives \(20\). For the denominators, \(4\) and \(5\) are multiplied to give \(20\) as well. The result is \(\frac{20}{20}\), which simplifies to \(1\). Multiplying fractions correctly is essential for getting the right answer in algebra problems. It can simplify complex expressions and is crucial for handling portions and ratios.
Simplification Steps in Algebra
Simplification is about making an expression easier to manage. In the given problem, we first distribute the terms, then multiply the fractions, and finally, simplify the result. Following each step methodically ensures that we don鈥檛 miss any detail and our final answer is accurate. Proper simplification transforms \-\frac{5}{4}\left(-\frac{4}{5} x\right)\ into \(x\). Breaking down each part makes this achievable. Remember, simplification isn鈥檛 just a mechanical process; it鈥檚 also about understanding what each step achieves and seeing the broader picture in algebra.

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