Problem 66 - Precalculus

Question

Explain the mistake that is made. Evaluate the function for the given \(x: f(x)=4^{x}\) for \(x=\frac{3}{2}\) Solution: \(\quad f\left(\frac{3}{2}\right)=4^{3 / 2}\) $$ =\frac{4^{3}}{4^{2}}=\frac{64}{16}=4 $$ The correct value is \(8 .\) What mistake was made?

Step-by-Step Solution

Answer

The mistake was incorrectly applying the rules of exponents by rewriting \( 4^{3/2} \) as \( \frac{4^3}{4^2} \) instead of using \( \left(4^{1/2}\right)^3 \).

Step 1: Understanding the Function

The function given is \( f(x) = 4^x \). To find \( f\left(\frac{3}{2}\right) \), we need to substitute \( x = \frac{3}{2} \) into the function, giving \( f\left(\frac{3}{2}\right) = 4^{\frac{3}{2}} \).

Step 2: Correct Calculation of the Expression

The expression \( 4^{\frac{3}{2}} \) means taking the square root of 4 and then raising it to the power of 3. The square root of 4 is 2, therefore \( 4^{\frac{3}{2}} = \left(\sqrt{4}\right)^3 = 2^3 = 8 \).

Step 3: Identify the Mistake

The mistake made in the original solution is applying the rules of exponent incorrectly. Instead of using \( 4^{3/2} = (4^{1/2})^3 \), they mistakenly split it into two separate powers \( \frac{4^3}{4^2} \), which is not equivalent to \( 4^{3/2} \).

Key Concepts

Fractional Exponents

Fractional exponents can sometimes seem confusing, but they are a helpful way of expressing powers and roots in a unified form. When you see an expression like \( a^{\frac{m}{n}} \), this represents the \(n\)-th root of \(a\) raised to the \(m\)-th power. For example, \( 4^{\frac{3}{2}} \) means you take the square root of 4 and then raise it to the power of 3. This translates into \( \left(\sqrt{4}\right)^3 \).

Understanding fractional exponents allows you to simplify complex expressions quickly. When approaching a problem involving fractional exponents, remember these steps:

• First, compute the root specified by the denominator.
• Then, apply the power indicated by the numerator.

This process can make seemingly tough problems more manageable, enabling you to find the correct value systematically.

Radicals

Radicals are another way of referring to roots in mathematics, with the square root being one of the most common types. Any expression written with a radical symbol \( \sqrt{} \) allows you to denote the square root.

In our example, we involved a square root when calculating \( 4^{\frac{3}{2}} \). To begin, we looked at \( \sqrt{4} \), which equals 2. Once the root calculation was done, you then raised this result to the power of 3, achieving \( 2^3 = 8 \). Understanding radicals will simplify many problems involving roots and exponents, enhancing your ability to perform calculations more effectively.

Always remember,

• Simplify the radical first if possible.
• Then perform any additional multiplication or exponentiation.

These steps ensure accuracy when working with radicals.

Mathematical Errors

Mathematical errors often occur when the rules of arithmetic are not applied correctly. In the original problem solution, a misinterpretation of how to handle fractional exponents led to the incorrect calculation. The mistake was treating \(4^{\frac{3}{2}}\) as \(\frac{4^3}{4^2} \). This is not a valid transformation.

To avoid such errors:

• Double-check that you're using the correct rule for the type of expression.
• Perform operations in the correct order.
• Use parentheses to clarify parts of your expressions where necessary.

By carefully following these guidelines, you can reduce the chances of mistakes and improve your problem-solving skills significantly. Precision is key in mathematics, and understanding where errors occur allows for more accurate answers and a deeper comprehension of the topic.