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Chapter 1: Problem 38

Evaluate each expression without using a calculator. $$ 16^{-3 / 4} $$

Short Answer

Expert verified
The evaluated expression is \( \frac{1}{8} \).

Step by step solution

01

Understanding the Negative Exponent

The expression involves a negative exponent, which means \[ 16^{-3/4} = \frac{1}{16^{3/4}} \]This step involves understanding that a negative exponent indicates taking the reciprocal of the positive exponent.
02

Simplify the Exponent Form

Now, we focus on simplifying \( 16^{3/4} \). The exponent \( 3/4 \) refers to the cube of the fourth root of 16. We can rewrite it as:\[ 16^{3/4} = (16^{1/4})^3 \] where \( 16^{1/4} \) denotes the fourth root of 16.
03

Calculate the Fourth Root

Calculate the fourth root of 16. Since 16 can be written as \( 2^4 \), we have:\[ 16^{1/4} = (2^4)^{1/4} = 2^{4/4} = 2 \]Thus, the fourth root of 16 is 2.
04

Cube the Fourth Root

Now, we cube the result from the previous step:\[ (16^{1/4})^3 = 2^3 = 8 \] This gives us the value of \( 16^{3/4} \) as 8.
05

Take the Reciprocal

Finally, use the result from Step 4 to calculate the reciprocal, which was derived in Step 1:\[ 16^{-3/4} = \frac{1}{8} \]This step concludes the calculations and provides the final answer to the expression.

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Key Concepts

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

Negative Exponents
When we encounter a negative exponent, it often looks intimidating, but it's just a different way to express fractions. A negative exponent tells us to take the reciprocal of the base raised to the corresponding positive exponent. For example, when you see \( 16^{-3/4} \), it simply means \( \frac{1}{16^{3/4}} \).
  • To deal with negative exponents, remember that the minus sign is there to tell you to "flip" the fraction.
  • This "flipping" process is similar whether the exponent is a whole number or a rational number.
By converting negative exponents into reciprocals, they become easier to manage in calculations.
Rational Exponents
Rational exponents, such as \( 16^{3/4} \), are another way of expressing roots and powers. The fraction in the exponent \( \frac{3}{4} \) means the fourth root of a number raised to the third power. This can be expressed as \( (16^{1/4})^3 \).
  • The denominator in the rational exponent indicates the root. In our example, 4 tells us we're taking the fourth root.
  • The numerator indicates the power. Here, 3 means that the fourth root result is to be cubed.
Using rational exponents often simplifies operations, especially when dealing with roots and powers together.
Simplifying Exponents
Simplifying expressions with exponents transforms complex problems into simpler ones. Continuing with \( 16^{3/4} \), we see this simplification is broken down into manageable steps:
  • First, you take the fourth root of 16, which means finding a number that, when multiplied by itself four times, gives 16. Knowing that \( 16 = 2^4 \), the fourth root is 2.
  • Then, cube this result (that is, multiply it by itself twice more): \( 2^3 = 8 \).
This step-by-step simplification helps in understanding not only the problem at hand but also builds confidence in tackling other exponential expressions.
Roots and Powers
Understanding the relationship between roots and powers is key to mastering exponents.
Any number raised to a power is a repeated multiplication, while roots are the opposite: finding what number can be multiplied several times to yield the original number.
In the expression \( 16^{3/4} \):
  • The fourth root: \( 16^{1/4} = 2 \) because \( 2^4 = 16 \).
  • The power: \( (2)^3 = 8 \) which completes the calculation.
Visualizing roots as finding the "building blocks" of a number and powers as "constructing" using those blocks can make these concepts more intuitive.

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Most popular questions from this chapter

BUSINESS: MBA Salaries Starting salaries in the United States for new recipients of MBA (master of business administration) degrees have been rising approximately linearly, from \(\$ 78,040\) in 2005 to \(\$ 89,200\) in \(2010 .\) a. Use the two (year, salary) data points \((0,78.0)\) and \((5,89.2)\) to find the linear relationship \(y=m x+b\) between \(x=\) years since 2005 and \(y=\) salary in thousands of dollars. b. Use your formula to predict a new MBA's salary in 2020 . [Hint: Since \(x\) is years after 2005, what \(x\) -value corresponds to \(2020 ?]\)

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=\frac{3}{x} $$

How do two graphs differ if their functions are the same except that the domain of one excludes some \(x\) -values from the domain of the other?

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=2 x^{2}-5 x+1 $$

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=\frac{1}{x^{2}} $$
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