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Chapter 1: Problem 35

Evaluate each expression without using a calculator. $$ 4^{-3 / 2} $$

Short Answer

Expert verified
\( 4^{-3/2} = \frac{1}{8} \).

Step by step solution

01

Understand the Expression

The expression given is \( 4^{-3/2} \). This is a power expression with a fractional and negative exponent, which indicates the involvement of square roots and reciprocals.
02

Rewrite the Negative Exponent

A negative exponent means we take the reciprocal of the base. Hence, \( 4^{-3/2} = \frac{1}{4^{3/2}} \). This step helps simplify the handling of exponents.
03

Evaluate the Fractional Exponent

The expression now is \( \frac{1}{4^{3/2}} \). A fractional exponent like \( 3/2 \) can be rewritten as a power and a root: \( 4^{3/2} = (\sqrt{4})^3 \).
04

Evaluate the Square Root

Find the square root of 4: \( \sqrt{4} = 2 \). This means \( (\sqrt{4})^3 \) becomes \( 2^3 \).
05

Evaluate the Remaining Exponent

Calculate \( 2^3 = 8 \). Therefore, the expression \( 4^{3/2} \) evaluates to 8.
06

Complete the Expression

Return to the reciprocal from Step 2: \( \frac{1}{4^{3/2}} = \frac{1}{8} \). This is the final evaluation of the expression.

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Key Concepts

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

Negative Exponent
Negative exponents can seem tricky at first, but they follow a simple rule. When you see a negative exponent, like in the expression \( a^{-n} \), it means you should take the reciprocal of the base raised to the corresponding positive exponent. In simpler terms, \( a^{-n} = \frac{1}{a^n} \).

This transformation involves flipping the base to the denominator (or numerator if it鈥檚 already in that form). For example, if you encounter \( 4^{-3/2} \), rewrite it as \( \frac{1}{4^{3/2}} \). This allows you to handle the rest of the expression without worrying about the negative sign.

Remember these key points when working with negative exponents:
  • A negative exponent indicates a reciprocal action.
  • Always convert the negative exponent into a positive form.
  • After flipping the base, continue to calculate as normal.
Fractional Exponent
Fractional exponents represent both roots and powers in a single expression. They can be split into two parts: the numerator and the denominator. The denominator introduces a root, while the numerator indicates the power. For instance, the expression \( a^{m/n} \) means the \( n \)-th root of \( a \) raised to the \( m \)-th power.

To simplify, take \( 4^{3/2} \) as an example. Here, the 2 in the denominator means you need the square root of 4, which is 2. The numerator, 3, tells you to raise that result to the third power: \( (\sqrt{4})^3 = 2^3 \). This process helps break down complex expressions step by step.

Some tips for working with fractional exponents:
  • Identify the root from the denominator.
  • Identify the power from the numerator.
  • Apply the root first, then the power, or vice versa, depending on the context.
Square Roots
Square roots are fundamental in mathematics, providing a way to find a number that, when multiplied by itself, results in the original number. The square root of a number \( x \) is often denoted as \( \sqrt{x} \).

To see this in action, consider the number 4. Its square root is 2, because \( 2 \cdot 2 = 4 \). Understanding square roots is essential when dealing with fractional exponents. In \( 4^{3/2} \), for example, you first find the square root of 4, which simplifies the expression significantly.

Key aspects to remember about square roots:
  • The square root represents a number that multiplied by itself gives the original number.
  • They are commonly used in physics, geometry, and algebra to simplify expressions.
  • When included in fractional exponents, they aid in breaking down complex expressions.

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Most popular questions from this chapter

BUSINESS: Isocost Lines An isocost line (iso means "same") shows the different combinations of labor and capital (the value of factory buildings, machinery, and so on) a company may buy for the same total cost. An isocost line has equation $$ w L+r K=\mathrm{C} \quad \text { for } L \geq 0, \quad K \geq 0 $$ where \(L\) is the units of labor costing \(w\) dollars per unit, \(K\) is the units of capital purchased at \(r\) dollars per unit, and \(C\) is the total cost. Since both \(L\) and \(K\) must be nonnegative, an isocost line is a line segment in just the first quadrant. a. Write the equation of the isocost line with \(w=8, \quad r=6, \quad C=15,000\), and graph it in the first quadrant. b. Verify that the following \((L, K)\) pairs all have the same total cost. \((1875,0),(1200,900),(600,1700),(0,2500)\)

GENERAL: Boiling Point At higher altitudes, water boils at lower temperatures. This is why at high altitudes foods must be boiled for longer times - the lower boiling point imparts less heat to the food. At an altitude of \(h\) thousand feet above sea level, water boils at a temperature of \(B(h)=-1.8 h+212\) degrees Fahrenheit. Find the altitude at which water boils at \(98.6\) degrees Fahrenheit. (Your answer will show that at a high enough altitude, water boils at normal body temperature. This is why airplane cabins must be pressurized - at high enough altitudes one's blood would boil.)

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

The intersection of an isocost line \(w L+r K=C\) and an isoquant curve \(K=a L^{b}\) (see pages 18 and 32 ) gives the amounts of labor \(L\) and capital \(K\) for fixed production and cost. Find the intersection point \((L, K)\) of each isocost and isoquant. [Hint: After substituting the second expression into the first, multiply through by \(L\) and factor.] $$ 3 L+8 K=48 \text { and } K=24 \cdot L^{-1} $$

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Slope \(-2.25\) and \(y\) -intercept 3
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