Problem 90 - Algebra and Trigonometry

Question

Evaluate each expression without using a calculator. $$ 16^{-\frac{5}{2}} $$

Step-by-Step Solution

Answer

So, the simplified form of \(16^{-5/2}\) is \(\frac{1}{1024}\).

Step 1: Definition of Negative Exponent

Firstly, we know that \(a^{-m} = \frac{1}{a^m}\). So, by using the definition of negative exponents, we can rewrite \(16^{-5/2}\) as \(\frac{1}{16^{5/2}}\).

Step 2: Simplification of Fractional Exponent

A number in the form \(a^{m/n}\) is equal to the \(n\)th root of \(a^m\). Hence, we can write \(16^{5/2} = (16^{1/2})^5\) which simplifies to \((4)^5\) (since the square root of 16 is 4).

Step 3: Calculate the Exponent

Now, we calculate the power of 5 on 4. This results in \(4^5 = 1024\). The original equation \(\frac{1}{16^{5/2}}\) now becomes \(\frac{1}{1024}\).

Key Concepts

Fractional Exponents

Fractional exponents might seem tricky at first, but they are a handy way to express roots and powers using a single format. When you see an exponent in the form of a fraction, like \(a^{m/n}\), it means you are dealing with both an exponent and a root at the same time. Here's how it works:

• The denominator (\(n\)) of the fraction represents the root you will take. So \(a^{1/n}\) is the \(n\)th root of \(a\).
• The numerator (\(m\)) signifies the power you will raise the result to after taking the root. Thus \(a^{m/n} = (a^{1/n})^m\).

For example, \(16^{1/2}\) asks you to find the square root of 16, which is 4. And when you multiply 4 with itself 5 times, you get \(4^5 = 1024\). That's how you handle fractional exponents to simplify expressions!

Simplifying Exponents

When simplifying exponents, particularly when you have negative or fractional exponents, it's important to follow a step-by-step process to make the expression more manageable. The key here is understanding and correctly applying the laws of exponents:

• Negative exponents involve taking the reciprocal of the base raised to the positive exponent: \(a^{-m} = \frac{1}{a^m}\).
• Fractional exponents follow the rule of roots and powers, as discussed before.

To simplify \(16^{-5/2}\), the negative exponent tells us to take the reciprocal, turning it into \(\frac{1}{16^{5/2}}\). Then, simplifying the fractional exponent involves identifying that \(16^{5/2}\) can be rewritten as \((16^{1/2})^5 = 4^5 = 1024\). Finally, the expression \(\frac{1}{16^{5/2}}\) becomes \(\frac{1}{1024}\).
By following these steps, you can break down complex expressions into simpler, solvable ones.

Radical Expressions

Radicals, or root expressions, are another way to represent fractional exponents. Notably, \(a^{1/n}\) can be expressed as \(\sqrt[n]{a}\), connecting directly to our earlier discussion on fractional exponents. These expressions often require simplifying to make them easier to work with:

• The square root (\(\sqrt{a}\)) is the most common radical, equivalent to raising \(a\) to the power of \(1/2\).
• Cubic roots, or \(\sqrt[3]{a}\), follow the same pattern but for the power \(1/3\).

In the exercise \(16^{5/2}\), converting to the radical form involves recognizing \((16^{1/2})\) as \((\sqrt{16})\), which simplifies to 4 due to the fact that the square root of 16 is 4. Raise it to the 5th power to get the final value within the reciprocal, \(\frac{1}{1024}\).
This showcases how radical expressions can make understanding the nature of these problems more intuitive, by physically representing the root involved.