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Chapter 1: Problem 5

Decide whether each statement is true or false. If it is false, correct the statement so that it is true. \((-6)^{7}\) is a negative number.

Short Answer

Expert verified
True, \((-6)^{7}\) is a negative number.

Step by step solution

01

Identify the Base and Exponent

The expression \((-6)^{7}\) consists of a base \(-6\) and an exponent \ 7\.
02

Determine Sign of the Result

An odd exponent applied to a negative number will result in a negative number. Therefore, \((-6)^{7}\) will be negative.
03

Validate Original Statement

The original statement is \((-6)^{7}\) is a negative number. Given that an odd exponent keeps the sign of the base, this statement is true.

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Key Concepts

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

Negative Exponents
Negative exponents can be confusing, but they follow simple rules. When you see a negative exponent, it means you take the reciprocal of the base. For instance, an expression like \( a^{-n} \) becomes \( \frac{1}{a^{n}} \). This flips the base to the denominator of a fraction and changes the exponent to positive.
Remember, only the exponent is changed to negative; the base remains the same. So, \(10^{-2} \) becomes \( \frac{1}{10^{2}} \), which is \( \frac{1}{100} \).
This reciprocal nature of negative exponents is handy in algebra, especially when simplifying expressions.
Odd Exponents
Odd exponents have specific effects when applied to negative bases. An exponent is odd if it is in the form of \(2n+1 \), where \( n \) is an integer.
When raising a negative base to an odd exponent, the result remains negative. For example, \[ (-3)^3 = -27 \] because the negative sign is multiplied an odd number of times, preserving the negative sign.
This is crucial in solving equations and understanding the behavior of functions. Recognizing that \[ (-6)^7 = -(6^7) \] and keeps the sign helps in verifying solutions, as demonstrated in the exercise above.
Sign Rules in Algebra
Understanding the sign rules in algebra is essential. These rules determine how you handle positive and negative signs in different operations.
Here are the basics:
  • Multiplying two positive numbers or two negative numbers gives a positive result.
  • Multiplying a positive number by a negative number gives a negative result.
  • Adding two positive numbers or two negative numbers depends on arithmetic signs.
  • Adding a positive number and a negative number requires considering their absolute values.

These sign rules extend to exponents. For example: applying a negative exponent inverses the base, while an even exponent results in a positive outcome regardless of the base sign.
In the case of the exercise, the odd exponent \(7 \) kept \((-6)^{7} \) negative, respecting the sign rules.

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Most popular questions from this chapter

An approximation of the average price of a theater ticket in the United States from 1977 through 2007 can be obtained by using the expression. $$ 0.1399 x-274.4 $$ where \(x\) represents the year. (Source: National Association of Theater Owners.) (a) Use the expression to complete the table. Round answers to the nearest cent. (b) How has the average price of a theater ticket in the United States changed from 1977 to \(2007 ?\)

The distributive property can be used to mentally perform calculations. For example, calcu- late \(38 \cdot 17+38 \cdot 3\) as follows. \(38 \cdot 17+38 \cdot 3=38(17+3) \quad\) Distributive property \(=38(20)\) Add inside the parentheses. \(=760\) Multiply. Use the distributive property to calculate each value mentally. $$ 58 \cdot \frac{3}{2}-8 \cdot \frac{3}{2} $$

Exercises \(91-116\) provide more practice on operations with fractions and decimals. Perform the indicated operations. $$\frac{8}{25}\left(-\frac{5}{12}\right)$$

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Solve each problem. Charlene Macdowall owes \(\$$237.59 on her MasterCard account. She returns one item costing \)\$$47.25 for credit and then makes two purchases of \(\$$12.39 and \)\$$20.00. (a) How much should her payment be if she wants to pay off the balance on the account? (b) Instead of paying off the balance, she makes a payment of \(\$$75.00 and incurs a finance charge of \)\$$32.06. What is the balance on her account
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