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Chapter 11: Problem 35

Evaluate each of the numerical expressions. $$-16^{\frac{5}{4}}$$

Short Answer

Expert verified
The expression has no solution in the real numbers.

Step by step solution

01

Understand the Expression

The expression to evaluate is \(-16^{\frac{5}{4}}\). Note that the negative sign is part of the base, so the expression can be interpreted as raising \(-16\) to the power of \(\frac{5}{4}\).
02

Break the Exponent

The exponent \(\frac{5}{4}\) can be split into the operation of first taking the fourth root of \(-16\) and then raising the result to the power of 5.
03

Evaluate the Fourth Root

Find the fourth root of \(-16\). In the real number system, the fourth root of a negative number is not defined because raising any real number to an even power results in a positive number.
04

Conclusion

Since the fourth root of \(-16\) is not defined within the real numbers, the expression \(-16^{\frac{5}{4}}\) does not have a real number solution.

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Key Concepts

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

Understanding Real Numbers
Real numbers are the set of numbers that include all the rational and irrational numbers. They form the basis of most calculations in algebra and represent all points on an infinitely long number line. Here are some key points about real numbers:
  • **Rational Numbers**: These can be expressed as fractions, like \( \frac{3}{4} \) or whole numbers like 5.
  • **Irrational Numbers**: These numbers cannot be expressed as simple fractions, such as \( \pi \) or \( \sqrt{2} \).
The real numbers do not include imaginary numbers. When dealing with roots, this differentiation becomes important, as certain roots of negative numbers (like the fourth root) don't exist within real numbers.
Demystifying Exponents
Exponents are a mathematical notation indicating the number of times a number is multiplied by itself. For instance, \( 3^4 \) means multiply 3 by itself four times, resulting in 81. Exponents follow specific rules:
  • **Product Rule**: \( a^m \times a^n = a^{m+n} \)
  • **Quotient Rule**: \( \frac{a^m}{a^n} = a^{m-n} \)
  • **Power of a Power Rule**: \( (a^m)^n = a^{m \cdot n} \)
When exponents are fractions, like \( \frac{5}{4} \), they indicate both a root and a power. \( -16^{\frac{5}{4}} \) involves both taking roots and using powers.
Understanding these core rules helps simplify and solve expressions correctly.
Exploring Roots in Mathematics
Roots are the inverse operation of exponents. The most common roots you encounter include the square root and cube root, but roots can be extended to any positive integer value. When dealing with roots:
  • The square root of a number \( a \) is a value \( b \) such that \( b^2 = a \).
  • The cube root of a number \( a \) is a value \( b \) such that \( b^3 = a \).
For even roots like the fourth root, you always need to be cautious with negative numbers, as raising a real number to an even power results in a positive number. This makes even roots of negative numbers undefined within the realm of real numbers, like the fourth root of \(-16\).
Recognizing when roots apply or do not apply in real numbers ensures the correct evaluation of expressions.

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Most popular questions from this chapter

For Problems \(35-54\), find each product and express it in the standard form of a complex number \((a+b i)\). $$ (-3-7 i)(-6-10 i) $$

For Problems \(35-54\), find each product and express it in the standard form of a complex number \((a+b i)\). $$ (4+2 i)(6+5 i) $$

The annual cost, in dollars, of $$\$ 100,000$$ of life insurance is a function of a person's age \(x\) and can be expressed as \(c(x)=354+10(x-30)\). Create a table showing the annual cost of insurance for a 20-year-old, a 30-yearold, a 35 -year-old, a 40-year-old, a 50-year-old, and a 55-year-old.

Most spreadsheet applications for computers allow you to enter a function, and when you highlight a list of values, the program evaluates the function for those values. If you have access to spreadsheet software, try evaluating the function \(f(x)=10 x+20\) for \(x=10,25,30,40\), and 50 .

The ABC Car Rental uses the function \(f(x)=26\) for the charges, in dollars, for the daily use of a car driven up to and including 200 miles. When a car is driven more than 200 miles per day, they use the function \(g(x)=26+0.15(x-200)\) to determine the charges. How much would they charge for daily driving of 150 miles, 230 miles, 360 miles, and 430 miles?
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