Problem 68 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\frac{4^{\frac{3}{2}}-16^{\frac{3}{2}}}{8^{\frac{1}{3}}}\)

Step-by-Step Solution

Answer

The simplified expression is -28.

Step 1: Simplify the Numerator - Format the Terms

The expression in the numerator is \(4^{\frac{3}{2}} - 16^{\frac{3}{2}}\). We can express the base numbers 4 and 16 as powers of 2: \(4 = 2^2\) and \(16 = 2^4\). Then rewrite the expression as \((2^2)^{\frac{3}{2}} - (2^4)^{\frac{3}{2}}\).

Step 2: Simplify the Powers in the Numerator

Using the property \((a^m)^n = a^{mn}\), simplify. For \((2^2)^{\frac{3}{2}}\), we get \(2^{2 \times \frac{3}{2}} = 2^3 = 8\). For \((2^4)^{\frac{3}{2}}\), we get \(2^{4 \times \frac{3}{2}} = 2^6 = 64\). The numerator simplifies to \(8 - 64\).

Step 3: Calculate the Numerator

Subtract the expressions: \(8 - 64 = -56\). So, the simplified numerator is \(-56\).

Step 4: Simplify the Denominator

The expression in the denominator is \(8^{\frac{1}{3}}\). Express 8 as a power of 2: \(8 = 2^3\). Then, rewrite it as \((2^3)^{\frac{1}{3}}\) and simplify using the power of a power property, \((a^m)^n = a^{mn}\), which gives \(2^{3 \times \frac{1}{3}} = 2^1 = 2\).

Step 5: Divide the Numerator by the Denominator

Now divide the simplified numerator \(-56\) by the simplified denominator \(2\):\[\frac{-56}{2} = -28\]

Step 6: Final Answer

The simplified expression is \(-28\). Therefore, \(\frac{4^{\frac{3}{2}} - 16^{\frac{3}{2}}}{8^{\frac{1}{3}}} = -28\).

Key Concepts

Understanding Numerator and Denominator

In mathematics, when you see a fraction, it is composed of two main parts: the numerator and the denominator.

• The numerator is the top part of the fraction, which in this case is the expression \(4^{\frac{3}{2}} - 16^{\frac{3}{2}}\).
• The denominator is the bottom part, which for this problem is \(8^{\frac{1}{3}}\).

When simplifying expressions involving exponents, it's essential to handle each part carefully. First, focus on simplifying the numerator and denominator separately.
Understanding these terms will help you comprehend the process of division once both are in their simplest forms. The ultimate goal is to resolve any operations within each part before performing the division.

Mastering Exponent Rules

Exponent rules are the backbone of simplifying expressions that contain powers. Here are a few crucial exponent rules that we used to solve this exercise:

• Power of a Power Rule: When you have something like \((a^m)^n\), you multiply the exponents together to get \(a^{mn}\). In our solution, this was used for both the numerator and the denominator. For instance, \((2^2)^{\frac{3}{2}}\) simplifies to \(2^3\).
• Subtraction of Exponents: Once you have simplified terms with powers, like \(2^3\) and \(2^6\), you can then perform basic arithmetic operations like subtraction (\(8 - 64\)).
• Evaluating Roots: The expression \(8^{\frac{1}{3}}\) is simplified by recognizing it as a cube root, resulting in \(2\).

Learning these rules and applying them step-by-step ensures that you can handle even complex algebraic expressions with confidence.

Simplifying Algebraic Expressions

Algebraic expressions can often look intimidating, but breaking them down into manageable steps makes the process much easier. In our exercise, we tackled an expression involving both exponents and fractions.

• Express Numbers as Powers: Begin by expressing base numbers like 4 and 16 in terms of common bases (e.g., powers of 2). This allowed us to rewrite \(4^{\frac{3}{2}}\) as \((2^2)^{\frac{3}{2}}\).
• Use Power Rules to Simplify: Apply the power of a power rule to simplify each term in the expression. This helped transform complex terms into more straightforward numbers like 8 and 64.
• Perform Arithmetic Operations: Simplify further by performing arithmetic operations (e.g., subtraction) when dealing with the terms 8 and 64 in the numerator.
• Simplify Entire Expression: Finally, divide the simplified numerator by the simplified denominator to achieve a simplified algebraic expression, \(-28\).

Approaching algebraic expressions with these strategies will help you simplify them effectively, making the problems more approachable.