Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$25^{-3 / 2}$$

Step-by-Step Solution

Answer

The value of \(25^{-3/2}\) is \(\frac{1}{125}\).

Step 1: Understand the Expression

The expression provided is \(25^{-3/2}\). This is in the form of \(a^{m/n}\) where \(a = 25\), \(m = -3\), and \(n = 2\).

Step 2: Convert to Radical Form

Using the property \(a^{m/n} = (a^{1/n})^m\), we can convert \(25^{-3/2}\) to \((25^{1/2})^{-3}\). Here, \(25^{1/2}\) is the square root of 25.

Step 3: Calculate the Square Root

Calculate \(25^{1/2}\), which is the square root of 25. The result is \(5\) because \(5^2 = 25\).

Step 4: Apply the Negative Exponent

Now apply the \(-3\) exponent to \(5\). So we calculate \((5)^{-3}\). A negative exponent indicates a reciprocal, so \((5)^{-3} = \frac{1}{5^3}\).

Step 5: Calculate the Power

Calculate \(5^3\), which is \(5 \times 5 \times 5 = 125\). So \((5)^{-3} = \frac{1}{125}\).

Key Concepts

Understanding Negative Exponents

In mathematics, a negative exponent represents the reciprocal of a number raised to the opposite positive exponent. This can sound complex at first, but with a bit of practice, it becomes easier to understand. Let's break it down:

• A negative exponent like \( a^{-m} \) is equivalent to \( \frac{1}{a^m} \).
• The negative sign simply indicates that you should take the reciprocal. For example, \((5)^{-3}\) becomes \(\frac{1}{5^3}\).

To grasp this deeply, imagine it as flipping the fraction. By understanding this concept, you can solve expressions with negative exponents by translating them into their reciprocal forms. In our exercise, after simplifying the expression inside the parentheses, we used the negative exponent to transform \((5)^{-3}\) into \(\frac{1}{125}\). This step is crucial to mastering negative exponents.

Exploring Radical Expressions

Radical expressions often appear in the form of square roots, cube roots, and so on. In our problem, we dealt with the square root of 25. Let's dive a bit deeper into this:

• A radical expression typically involves a root, symbolized by the radical sign \(\sqrt{} \).
• For example, \(25^{1/2}\) or \(\sqrt{25}\) signifies the square root of 25.
• Being a perfect square, 25 simplifies neatly to 5, which makes \(\sqrt{25} = 5\).

The concept here hinges on finding the number which, when multiplied by itself, gives the original number. Understanding radical expressions is essential as they frequently appear in various mathematical scenarios and are important foundational skills.

Grasping the Power of a Number

The power of a number involves multiplying the number by itself a specific number of times, determined by the exponent. This is often referred to as "raising" a number to a certain power:

• For example, \(5^3\) means \(5\times5\times5\).
• This results in 125, as demonstrated in our step-by-step solution.

Understanding how to compute the power of a number is vital when dealing with exponents. Whether you're working with positive or negative exponents, or even resolving complex expressions, knowing how to calculate powers efficiently forms the basis for success in many mathematical topics.