Problem 132 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I use the definition for \(a^{\frac{m}{n}}\) I usually prefer lo first raise \(a\) to the \(m\) power because smaller numbers are involved.

Step-by-Step Solution

Answer

The statement does not make sense because raising \(a\) to the \(m\) power first does not always result in simpler calculations, especially when \(a\) or \(m\) are large.

Step 1: Understanding the statement

First, let's understand what the statement is arguing. It suggests that when dealing with a number raised to a fractional power, like \(a^{\frac{m}{n}}\), it seems easier to first raise \(a\) to the power of \(m\) and then take the \(n\)th root, because it involves smaller numbers.

Step 2: Analyze the operation

Now let's scrutinize this approach by taking a simple example such as \(a=2, m=3, n=2\). Following the statement's suggestion, first calculate \(2^3 = 8\), then find the square root of 8, which gives approximately \(2.83\). However, if we take into account the definition of fractional powers, which is to first take the \(n\)th root of \(a\) and then raise it to the \(m\)th power, the operation would be: \(\sqrt[2]{2} = 1.414\), then \(1.414^3 \approx 2.83\). As we can see, both approaches generate the same result. However, the highlighted 'smaller numbers' advantage doesn't always apply. If \(m\) were significantly larger than \(n\), for instance, calculating \(a^m\) could result in a very large number, making it harder to compute the root.

Step 3: Evaluate the validity of the statement

Therefore, the statement doesn't hold up under scrutiny. Although there doesn't seem to be any disadvantage to following this method with smaller exponents, it isn't a universal truth that smaller numbers will be involved if \(a\) is raised to the \(m\)th power first. The larger \(a\) and \(m\) are, the more complicated the calculations will become with this approach.

Key Concepts

Fractional Exponents

The concept of fractional exponents can seem tricky at first. It's essentially a way to express roots using powers. Let’s break down the expression \( a^{\frac{m}{n}} \). Here, \( a \) is the base, while \( \frac{m}{n} \) is the fractional exponent.
- **Between Fractions and Exponents**: The denominator \( n \) represents the root, whereas the numerator \( m \) signifies the exponent. So, \( a^{\frac{m}{n}} \) tells us to take the \( n \)th root of \( a \) and then raise the result to the \( m \)th power. This can be written as \( (\sqrt[n]{a})^m \).
Understanding fractional exponents is crucial because it's a more general form to handle expressions involving roots and powers efficiently. This helps in simplifying and solving equations involving non-integer powers.

Roots and Powers

When dealing with roots and powers, it's important to understand how they can be interchangeable in expressions like \( a^{\frac{m}{n}} \).
- **Roots**: The \( n \)th root of \( a \), \( \sqrt[n]{a} \), is the number that multiplied by itself \( n \) times will give \( a \).
- **Powers**: Raising \( a \) to the \( m \)th power, written as \( a^m \), means multiplying \( a \) by itself \( m \) times.
Given these definitions, when you see a fractional exponent, you’re using both roots and powers. For instance, \( a^{\frac{3}{2}} \) involves both taking the square root of \( a \), \( \sqrt{a} \), and then cubing it, or \( (\sqrt{a})^3 \). The order in which you perform these operations can depend on the size of \( m \) and \( n \) or personal preference since the result remains the same.

Mathematical Reasoning

Mathematical reasoning involves evaluating different methods and choosing the one that works best for the problem at hand. With fractional exponents, understanding the reasoning behind the operation is crucial.
- **Choosing an Approach**: When deciding whether to take a root first or to raise to the power first, consider the scale of the numbers involved. Taking the root first is often advised because it prevents the moments when large powers can result in cumbersome numbers that make calculations difficult.
- **Example Evaluation**: Suppose you have \( a = 2 \), \( m = 3 \), and \( n = 2 \). You can calculate \[ (\sqrt{2})^3 \] almost as easily as \[ \sqrt{8} \] after computing \( 2^3 \). Both methods give \( 2.83 \). But, establishing which path to take relies on numerical convenience.
Applying mathematical reasoning allows you to choose methods that lead to simpler calculations while still yielding the correct result.