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Chapter 1: Problem 46

Evaluate the given expressions. All numbers are approximate. $$-0.711^{2}-(-|-0.809|)^{6}$$

Short Answer

Expert verified
The value of the expression is approximately 0.245377.

Step by step solution

01

Evaluate Absolute Value

The expression contains an absolute value: \(-|-0.809|\). First, find the absolute value of -0.809, which is 0.809. Thus, \(|-0.809| = 0.809\). Then, apply the negative sign in front of the absolute value: \(-|-0.809| = -0.809\).
02

Evaluate Exponents

The expression includes squaring and raising to the sixth power. Calculate these powers as follows:1. Square -0.711: \((-0.711)^2 = 0.711^2 = 0.505521\).2. Raise -0.809 to the power of 6: \((-0.809)^6 = 0.809^6 = 0.260144\).Notice that when a negative number is raised to an even power, the result is positive.
03

Simplify the Expression

Write the equation by substituting the calculated powers into the original expression:\( -0.711^2 - (-|-0.809|)^6 = 0.505521 - 0.260144\).Subtract 0.260144 from 0.505521 to get:\(0.505521 - 0.260144 = 0.245377\).
04

Compile the Result

The simplified expression is now calculated. Therefore, \(-0.711^2 - (-|-0.809|)^6 = 0.245377\).

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Key Concepts

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

Understanding Absolute Value
Absolute value is a way to express how far a number is from zero on a number line, disregarding its direction. Think of it as a number's "magnitude without direction." When we find the absolute value of a number, we drop any negative sign.For instance, the absolute value of -0.809 is 0.809 because it is 0.809 units away from zero on the number line. Similarly, the absolute value of 0.809 is also 0.809, as it is already positive.Key things to remember:
  • Absolute value is always non-negative.
  • The symbol for absolute value is two vertical bars, like this: \(|x|\).
  • In expressions, always resolve absolute values first before applying other operations.
Dealing with Negative Numbers
Negative numbers are simply numbers with a value less than zero. They usually have a minus sign in front of them. But when working with expressions, particularly those involving absolute values and exponents, understanding how negative numbers behave is important.In the exercise, \(-|-0.809|\) means that after calculating the absolute value, we apply a negative sign to the result. So, whether a number is negative or positive can drastically change the outcome of mathematical expressions.Important points include:
  • Negative numbers are located to the left of zero on the number line.
  • They are often used to represent loss, debt, or decrease.
  • When raised to an even power, negative numbers become positive. This is a crucial transition in our calculations.
Effects of Even Powers
When raising numbers to the power of an even number, the result is always non-negative. This is because the negative signs cancel each other out.Let's look at the example of \((-0.711)^2\). Squaring means multiplying the number by itself. So, \(-0.711 \times -0.711\) results in a positive value because:
  • The two negative signs (- and-) combine to produce a positive product.
  • \((-0.711)^2\) simplifies to \(0.505521\).
Similarly, raising any negative number to an even power, like six in the case of \((-0.809)^6\), offers a positive outcome. This property simplifies many algebraic expressions and is crucial in calculations involving higher powers.

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Most popular questions from this chapter

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