Problem 1 - Intermediate Algebra

Question

The expressions \(4^{1 / 2}\) and \((-8)^{-2 / 3}\) have __________ exponents.

Step-by-Step Solution

Answer

fractional but of different signs.

Step 1: Understand the Exponent Types

The expression for an exponent in the form \(a^{b}\) where \(a\) is the base and \(b\) is the exponent. Here, for \(4^{1/2}\), the base is 4 and the exponent is \(\frac{1}{2}\). For \((-8)^{-2/3}\), the base is -8 and the exponent is \(-\frac{2}{3}\).

Step 2: Identify Types of Exponents

Next, we identify the type of exponents involved: \(\frac{1}{2}\) is a positive fractional exponent, while \(-\frac{2}{3}\) is a negative fractional exponent. Both are non-integer exponents.

Step 3: Determine if Exponents are of the Same Type

Given that \(4^{1/2}\) uses a positive fractional exponent and \((-8)^{-2/3}\) uses a negative fractional exponent, the exponents are both fractional but different in sign.

Key Concepts

Understanding Positive Fractional Exponents

Fractional exponents provide a powerful way to represent roots. When you see something like a positive fractional exponent, it means taking a root of the number. For example, the expression \(4^{1/2}\) might look complex, but it really just asks us to find the square root of 4. That's right, \(4^{1/2} = \sqrt{4}\), which equals 2.

This concept becomes easier when you remember the following points:

• The numerator of the fractional exponent acts as a traditional exponent (power).
• The denominator indicates the root you need to extract. For instance, \(x^{1/3}\) is the cube root of \(x\).

So when you encounter a positive fractional exponent, just break it down into these steps to simplify your work.

Exploring Negative Fractional Exponents

Negative fractional exponents may seem intimidating, but they can be deciphered with a little practice. Consider the expression \((-8)^{-2/3}\). The negative sign in the exponent means you'll have to take the reciprocal of the base.

Here's how you break it down:

• First deal with the negative sign by finding the reciprocal. Change \((-8)^{-2/3}\) into \([\frac{1}{(-8)^{2/3}}]\).
• Next, handle the fractional part just like you would with a positive fractional exponent. Compute \((-8)^{2/3}\).
  The number 2 is a power, and 3 represents the cube root, so find the cube root of -8, which is -2 (since \(-2^3 = -8\)).
• Finally, square the result: \((-2)^2 = 4\), leading to \([\frac{1}{4}]\).

Negative fractional exponents simplify calculations involving roots and reciprocals!

Properties of Exponents

Understanding the properties of exponents is crucial for dealing with complex expressions. These properties help simplify mathematical operations and expressions involving exponents. Here are the key properties to remember:

• Product of Powers Property: \(a^m \times a^n = a^{m+n}\). This property helps when multiplying like bases.
• Quotient of Powers Property: \(a^m / a^n = a^{m-n}\). Use this when dividing like bases.
• Power of a Power Property: \((a^m)^n = a^{m \times n}\). Apply this when raising a power to another power.
• Zero Exponent Property: \(a^0 = 1\), indicating any non-zero number raised to the power of zero equals 1.
• Negative Exponent Property: \(a^{-m} = \frac{1}{a^m}\). This property is crucial for dealing with negative exponents effectively.

These properties can simplify exponential expressions and make math tasks more manageable. Practicing these makes handling exponents a breeze!