Problem 55 - A Graphical Approach to College Algebra

Question

Consider the expression \(16^{-3 / 4}\) (a) Simplify this expression without using a calculator. Give the answer in both decimal and \(\frac{a}{b}\) form. (b) Write two different radical expressions that are equivalent to it, and use your calculator to evaluate them to show that the result is the same as the decimal form you found in part (a). (c) If your calculator has the capability to convert decimal numbers to fractions, use it to verify your results in part (a).

Step-by-Step Solution

Answer

(a) Fraction: \(\frac{1}{8}\), Decimal: \(0.125\). (b) Radical forms: \(\frac{1}{(\sqrt[4]{16})^3}\) or \(\frac{1}{\sqrt[4]{16^3}}\); both equal 0.125. (c) Calculator confirms \(\frac{1}{8}\).

Step 1: Understanding Exponentiation

The given expression is \(16^{-3/4}\). The negative exponent indicates a reciprocal, so \(16^{-3/4} = \frac{1}{16^{3/4}}\). Our task in this step is to simplify \(16^{3/4}\).

Step 2: Simplifying Using Roots and Powers

Recognize that \(16^{3/4}\) involves both a fourth root and a cube. Rewrite it as \((16^{1/4})^3\). Notice that \(16 = 2^4\), so \(16^{1/4} = 2^1 = 2\). So \((16^{1/4})^3 = 2^3 = 8\). Thus, \(16^{3/4} = 8\).

Step 3: Reciprocal and Decimal Conversion

Now calculate \(16^{-3/4} = \frac{1}{8}\), which is already in its simplest fractional form. Converting \(\frac{1}{8}\) to a decimal gives \(0.125\).

Step 4: Writing Two Equivalent Radical Expressions

The expression \(16^{-3/4}\) can be written in radical form as \(\frac{1}{(\sqrt[4]{16})^3}\) or equivalently \(\frac{1}{\sqrt[4]{16^3}}\). Evaluating \(\sqrt[4]{16} = 2\) and calculating \((\sqrt[4]{16})^3 = 8\) gives \(\frac{1}{8} = 0.125\). Similarly, calculating \(\sqrt[4]{16^3} = \sqrt[4]{4096} = 8\) also gives \(\frac{1}{8} = 0.125\).

Step 5: Verification Using Calculator

Using a calculator, convert the decimal \(0.125\) back to a fraction. The calculator should confirm that \(0.125 = \frac{1}{8}\), verifying our earlier calculations.

Key Concepts

Radical Expression

A radical expression involves roots, such as square roots or cube roots. When we talk about radical expressions, we use the radical symbol \( \sqrt{} \) to denote them. For instance, \( \sqrt{4} \) means "the square root of 4," which is 2. In our exercise, to break down \( 16^{-3/4} \) using a radical expression, we rewrite the base as operations involving roots.
\[ 16^{-3/4} = \frac{1}{(\sqrt[4]{16})^3} \]
This expression shows us how to "unpack" the fractional exponent \(-3/4\). Let's go step by step to understand this better.

• Fractional Exponent: A fractional exponent such as \(3/4\) means both a power and a root action. Here, we see \(3/4\) which can be broken down to first find the fourth root (\(1/4\)) of 16 and then cubing \((3)\) that result.
• Taking the Fourth Root: Compute \(\sqrt[4]{16}\). We know that \(\sqrt[4]{16}\) equals 2 because 2 raised to the 4th power gives us back 16.

Altogether, we can express the calculation as \( (\sqrt[4]{16})^3 = 2^3 = 8 \), which then translates the entire radical into the reciprocal \( \frac{1}{8} \), a form that leads directly to our decimal \(0.125\) upon conversion.

Fraction to Decimal Conversion

Converting fractions to decimals is a fundamental skill in mathematics. The process involves division. For example, to convert \(\frac{1}{8}\) into a decimal, we divide 1 by 8.
A fraction like \(\frac{1}{8}\) means one piece of something divided into 8 equal pieces. When we perform this division,
\[ 1 \div 8 = 0.125 \]
This is the decimal representation of the fraction and is crucial for many applications, particularly when precise or comparative values are needed.

• Division Process: The division can be thought of as seeing how many times 8 fits into 1, considering carries and resulting quotient step by step.
• Accuracy: Decimals are often more immediate for calculations in practical scenarios, like when using a calculator, ensuring numerical accuracy in representation.

Fraction to decimal conversion aids in verifying computations within exercises, as it provides an alternative representation that can easily be cross-checked for equivalency.

Simplifying Expressions

Simplifying expressions, especially those involving powers and roots, helps streamline complex operations like being able to easily see the simplest form of a number or expression. In cases like \( 16^{-3/4} \), simplification involves understanding deeper steps beyond just calculating the answer.
Simplification is not just "calculating," it involves:

• Breaking down Composite Steps: Seeing \(16\) as \(2^4\) sheds light on further actions like taking specific roots based on exponents.
• Combining Rules of Exponents: The expression \(16^{-3/4} = \frac{1}{16^{3/4}} = \frac{1}{8}\) illustrates negative exponents, which essentially render a reciprocal, while fractional exponents handle roots.

The process of simplification confirms consistency across operational paths—both through direct exponentiation and radical manipulation—as each results in the same workable form \(\frac{1}{8}\) or decimal \(0.125\). By practicing simplification, one gains not only speed but an intuitive understanding of algebraic structures.