Problem 55 - A Graphical Approach to Precalculus with Limits

Question

Answer each of the following. Consider the expression \(16^{-34}\) (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) \(\frac{1}{2^{136}}\) and approximately \(1.135 \times 10^{-41}\). (b) Radicals: \((\sqrt{16})^{-68}\), \(\left(16^{\frac{1}{4}} \right)^{-136}\). (c) Verify using calculator to decimal \(1.135 \times 10^{-41}\).

Step 1: Simplifying the Expression

To simplify the expression \(16^{-34}\), recall that a negative exponent \(n^{-m}\) means \(\frac{1}{n^{m}}\). So, \(16^{-34} = \frac{1}{16^{34}}\). Since 16 is \(2^4\), rewrite it as \(16^{34} = (2^4)^{34} = 2^{136}\). Hence, \(16^{-34} = \frac{1}{2^{136}}\).

Step 2: Decimal Form of the Expression

The decimal form of \(\frac{1}{2^{136}}\) is extremely small because \(2^{136}\) is a very large number. Calculating \(2^{136}\) using a calculator gives \(87112285931760246646623899502532662132736\). So, \(\frac{1}{2^{136}} \approx 1.135 \times 10^{-41}\).

Step 3: Radical Expressions

Recall that radical expressions can be written using exponents as \(a^{\frac{1}{n}}\). Therefore, to find radicals equivalent to \(16^{-34}\), consider \((\sqrt{16})^{-68}\) or \(\left(16^{\frac{1}{4}} \right)^{-136}\). Both simplify to \(16^{-34}\). Use your calculator to check that these equal \(1.135 \times 10^{-41}\).

Step 4: Verification Using Calculator

Using a calculator, convert the decimal \(1.135 \times 10^{-41}\) back to a fraction. This confirms \(\frac{1}{2^{136}}\) since it's a power of 2.

Key Concepts

Negative exponents

When dealing with negative exponents, a common misconception is that they simply turn a number negative. Instead, a negative exponent indicates reciprocal behavior. For instance, in the expression \( n^{-m} \), the negative exponent \(-m\) means you take the reciprocal of \( n^m \). In simple terms, \( n^{-m} \) can be rewritten as \( \frac{1}{n^m} \).

• Think of it this way: with negative exponents, "flip" the base to the denominator.
• This flipping converts the negative exponent into a problem of division rather than multiplication.

For example, in the original exercise, \( 16^{-34} \) is equivalent to \( \frac{1}{16^{34}} \). Hence, understanding negative exponents is crucial for simplifying expressions without a calculator by seeing the inverse relationship they create.

Radical expressions

Radical expressions use roots to indicate a different form of exponentiation. The roots are expressed in fraction terms, such as \( a^{\frac{1}{n}} \), which can be interpreted as "the \(n\)-th root of \(a\)."

• For instance, \( \sqrt{16} = 16^{\frac{1}{2}} \), since squaring \(4\) gives \(16\).
• Radicals are handy for expressing large exponential expressions in a more compact form.

In the exercise, finding equivalents for \( 16^{-34} \) involves writing radical expressions like \( (\sqrt{16})^{-68} \). This matches with \( 16^{-34} \) due to exponent properties where multiplying the exponents inside and outside a radical (\(a^{b \cdot c}\)) results in the same power (using the property \( (a^b)^c = a^{b \times c} \)). Such expressions help verify results when using a calculator, ensuring consistency across different methods of simplification.

Decimal to fraction conversion

Converting decimals into fractions is a pivotal skill in mathematics, providing exact representations of numerical values. Why? Because fractions can express exact values that decimals approximate.

• A regular method includes utilizing your calculator's functionalities to convert a decimal result back into a fraction form.
• Manual conversion involves understanding the decimal place value (e.g., 0.5 means \( \frac{5}{10} \)).

In the exercise, the very small decimal \( 1.135 \times 10^{-41} \), derived from \( \frac{1}{2^{136}} \), can be challenging to directly convert without a calculator due to its size. However, knowing the process reinforces your understanding of how fractions and decimals can interchange, providing preciseness where needed. Decimal to fraction conversion also allows for easy verification, ensuring the mathematical operations lead to consistent results.