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Chapter 4: Problem 24

Simplify. $$-\left(5 \cdot \frac{2}{5}\right)^{2}$$

Short Answer

Expert verified
-4

Step by step solution

01

Simplify the expression inside the parentheses

First, simplify the expression inside the parentheses: \(5 \times \frac{2}{5}\). Since multiplication is commutative, this simplifies as follows: \(5 \times \frac{2}{5} = \frac{5 \times 2}{5} = 2.\)
02

Apply the exponent

Next, apply the exponent: \(2^2.\) When you square 2, you calculate: \(2^2 = 4.\)
03

Apply the negative sign

Finally, apply the negative sign in front of the expression: \(-\left(2^2\right).\) Thus, simplifying this expression gives: \(-4.\)

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Key Concepts

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

Prealgebra
Prealgebra is the foundation of mathematics that prepares you for algebra. It covers basic arithmetic operations and introduces fundamental concepts you鈥檒l need for more advanced math.

For example, in this exercise, we simplified the expression inside the parentheses by using basic multiplication. This step involves understanding how to handle fractions and integers. Prealgebra ensures you are comfortable with numbers and signs before moving on to more complex topics.
Exponents
Exponents represent how many times a number, called the base, is multiplied by itself. They are written as a small number to the upper right of the base.

In our expression, we have the term \(2^2\). This means we multiply 2 by itself: \(2 \times 2 = 4\). Understanding exponents is crucial as they frequently appear in algebra and higher-level mathematics.

Remember that an exponent only affects the number it is attached to. So always pay close attention to the placement of exponents in expressions.
Negative Numbers
Negative numbers are numbers less than zero. They are often used in various math problems and have distinct rules when doing arithmetic.

In the given problem, after simplifying inside the parentheses and calculating the exponent, we apply the negative sign in front of the final result. This changes the expression to its negative form. So, if you have \(4\) and you apply a negative sign, it becomes \(-4\).

Always remember that a negative sign in front of a whole expression means you need to change the sign of the result of that expression.
Commutative Property
The commutative property states that the order in which you add or multiply numbers does not change the result.

For example, \(5 \times \frac{2}{5} = \frac{2}{5} \times 5\). In our initial step, this property allows us to simplify the expression inside the parentheses regardless of how the numbers are ordered.

Understanding the commutative property is fundamental for simplifying expressions and for more advanced topics in mathematics, where changing the order of operations can often make problems easier to solve.

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Most popular questions from this chapter

Which expressions are equivalent to \(-\frac{4}{5} ?\) \(\begin{array}{lll}\text { a. } \frac{-4}{5} & \text { b. } \frac{-4}{-5} & \text { c. } \frac{4}{-5} & \text { d. }-\frac{5}{4}\end{array}\)

Simplify the fraction to lowest terms. Write the answer as a fraction or a whole number. $$\frac{84}{126}$$

Simplify. $$\left|-\frac{8}{7}\right|$$

Convert the improper fraction to a mixed number. $$-\frac{43}{18}$$

For Exercises 95鈥�102, simplify. $$\left|-\frac{3}{4}\right|$$
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