Problem 68 - College Algebra

Question

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: Convert Base Numbers

Start by expressing each base number as a power of 2. We know that \(4 = 2^2\), \(16 = 2^4\), and \(8 = 2^3\).

Step 2: Apply Exponents to Base Numbers

Apply the exponents to each base number: \(4^{\frac{3}{2}} = (2^2)^{\frac{3}{2}} = 2^{3}\), \(16^{\frac{3}{2}} = (2^4)^{\frac{3}{2}} = 2^{6}\), and \(8^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} = 2^{1}\).

Step 3: Simplify the Subtraction in the Numerator

In the expression \(2^3 - 2^6\), factor out \(2^3\). This gives us \(2^3 (1 - 2^3)\) which simplifies to \(2^3 (-7)\).

Step 4: Simplify the Overall Fraction

Now, simplify the fraction: \(\frac{2^3 (-7)}{2^1} = 2^{2} (-7) = -7 \times 4 = -28\).

Key Concepts

Fraction Simplification

Fraction simplification is a process of making a fraction as simple as possible, making the numerator and the denominator as small as they can be. In our problem, we are dealing with a complex-looking fraction:\[\frac{4^{\frac{3}{2}}-16^{\frac{3}{2}}}{8^{\frac{1}{3}}}\]To simplify, we first convert the bases to a common power using the properties of exponents. This helps us manage the expressions better. Simplification often involves:

• Finding common factors to reduce the fraction
• Using exponent rules to rewrite terms for easier calculation

Simplicity can sometimes reveal results you can compute easily, such as in our example where reducing the components allowed solving the operation swiftly. The essence of simplification lies in its ability to transform complicated expressions into more manageable forms by reducing terms to their simplest state.

Power of a Number

A power of a number refers to the product of multiplying the number by itself a certain number of times. The power consists of a base and an exponent. In the expression \(a^b\), "a" is the base, and "b" is the exponent. In this exercise, we demonstrated using powers by expressing base numbers as powers of 2:- For 4, we know it equals \(2^2\)- For 16, it equates to \(2^4\)- And for 8, it is \(2^3\) Using these powers, we can better manipulate the expression to simplify it. Calculating a power, like \(4^{\frac{3}{2}}\), involves applying exponents to the base power. Here, it was essential to convert each base to a power of 2, making the simplification of other expressions, like their subtraction or division, much easier. Hence, understanding how to express numbers as powers greatly aids in managing otherwise complex calculations. This powerful technique can reduce both computational time and error.

Factoring Exponents

Factoring exponents is an effective method when simplifying expressions that involve subtraction or addition of powers. In our exercise, one of the crucial steps in simplification was the factoring in the numerator.When looking at terms like \(2^3 - 2^6\), factoring helps by taking the smaller power out as a common term. Here,\(2^3\) was factored:

• Extract the common factor: \(2^3 (1 - 2^3)\)
• This changes the expression to a more simple form, highlighting any arithmetic operation needed - in this case, \(-7\)

This approach reduces complexity by focusing on multipliers and exposing underlying simple operations. The ability to factor common powers is an important skill that enhances understanding and application of exponent rules, consolidating one's grasp on algebraic operations.