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Chapter 4: Problem 10

Evaluate the expression by hand. $$ \frac{4^{-1 / 2}}{4^{3 / 2}} $$

Short Answer

Expert verified
The value of the expression is \( \frac{1}{16} \).

Step by step solution

01

Write the expression in terms of exponents

The given expression is \( \frac{4^{-1/2}}{4^{3/2}} \). It can be rewritten using the properties of exponents as \( 4^{-1/2} \cdot 4^{-3/2} \).
02

Combine exponents with the same base

According to the property \( a^m \cdot a^n = a^{m+n} \), we add the exponents.\[ 4^{-1/2 - 3/2} = 4^{-4/2} = 4^{-2} \]
03

Simplify the exponent

The expression \( 4^{-2} \) implies \( \frac{1}{4^2} \). Calculate \( 4^2 \):\[ 4^2 = 16 \] Thus, \( 4^{-2} = \frac{1}{16} \).
04

Find the value of the expression

The given expression simplifies to \( \frac{1}{16} \). Thus, the evaluated result of the expression is \( \frac{1}{16} \).

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Key Concepts

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

Understanding the Properties of Exponents
When working with exponents, there are several properties that make handling these expressions easier. One important property is the multiplication property, which states that when you multiply two powers with the same base you can add their exponents together. Mathematically, this is expressed as \( a^m \cdot a^n = a^{m+n} \). This property was used in the exercise solution to combine \( 4^{-1/2} \cdot 4^{-3/2} \).
By applying this property, the expression simplifies to \( 4^{(-1/2) + (-3/2)} = 4^{-4/2} \), which is equivalent to \( 4^{-2} \). This demonstrates how exponents can be managed systematically.
  • Exponents can be multiplied by adding their powers when sharing the same base.
  • This property is essential for simplifying expressions involving exponents.
  • Remember, the base must be the same for this property to apply.
How to Deal with Negative Exponents
Negative exponents often confuse students, but they're more straightforward if you understand what they represent. A negative exponent suggests the reciprocal of the base raised to the absolute value of the exponent. In other words, \( a^{-n} = \frac{1}{a^n} \).
In our exercise, we see this demonstrated when solving \( 4^{-2} \). The negative exponent tells us to take the reciprocal, giving us \( \frac{1}{4^2} \).
  • Negative exponents mean taking the reciprocal of the base.
  • \( a^{-n} \) becomes \( \frac{1}{a^n} \), significantly altering the expression.
  • This changes a potentially complex looking problem into a simple division.
Understanding negative exponents transforms daunting equations into manageable fractions.
Simplifying Expressions with Exponents
Simplifying expressions with exponents requires conflating all parts into their simplest forms. This often involves applying rules like those discussed for properties of exponents and handling negatives.
In our solved problem, we identified the same base in \( 4^{-1/2} \) and \( 4^{-3/2} \), combined them into one exponent, and simplified it by recognizing it as \( 4^{-2} \).
Then, utilizing the rule for negative exponents, we found \( 4^{-2} = \frac{1}{16} \).
  • Start by simplifying exponents wherever possible.
  • Combine like bases using exponent properties.
  • Always convert negative exponents using their reciprocal form.
This methodical approach to simplifying ensures clarity and precision, delivering the final answer: \( \frac{1}{16} \).

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Most popular questions from this chapter

Let \(f(x)\) be the formula for a rational function. (a) Explain how to find any vertical or horizontal asymptotes of the graph of \(f\) (b) Discuss what a horizontal asymptote represents.

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