Problem 98 - College Algebra Essentials

Question

Factor and simplify each algebraic expression. $$\left(x^{2}+4\right)^{\frac{3}{2}}+\left(x^{2}+4\right)^{\frac{7}{2}}$$

Step-by-Step Solution

Answer

The factored and simplified form of the given algebraic expression is \( \left(x^{2}+4\right)^{\frac{3}{2}} \cdot \left( 1+\left(x^{2}+4\right)^{2} \right) \).

Step 1: Identifying the Common Base

In the given exercise, the common base is \( \left(x^{2}+4\right) \). Both terms are raised to a power - the first term is raised to the power of \( \frac{3}{2} \) and the second term is raised to the power \( \frac{7}{2} \). This allows us to factor the expression.

Step 2: Factor the Expression

We can factor out \( \left(x^{2}+4\right)^{\frac{3}{2}} \) from both the terms. Factoring gives us the expression: \( \left(x^{2}+4\right)^{\frac{3}{2}} \cdot \left( 1+\left(x^{2}+4\right)^{2} \right) \).

Step 3: Final Simplified Form

The factored and simplified expression is \( \left(x^{2}+4\right)^{\frac{3}{2}} \cdot \left( 1+\left(x^{2}+4\right)^{2} \right) \). This is the final simplified form of the given algebraic expression.

Key Concepts

Common Base in Algebra

Understanding the concept of a common base is crucial when working with algebraic expressions that involve powers. It is akin to finding a repeating element that can be factored out to simplify an expression. When you come across terms in an expression that share a common base but are raised to different exponents, like in the exercise \( (x^{2}+4)^{\frac{3}{2}} + (x^{2}+4)^{\frac{7}{2}} \) , you're in a good position to factor and simplify.

Identifying and factoring out the common base allows you to combine terms in a way that is often more concise and easier to work with. Here, the common base is \( x^{2}+4 \) and by factoring out the lowest exponent, you've efficiently set the stage for further simplification. This concept is a fundamental strategy in algebra that leads to a clearer and more elegant solution.

Simplifying Expressions

Simplifying expressions is the process of reducing an algebraic expression to its simplest form. This means combining like terms, factoring out common elements, and performing arithmetic operations to reduce complexity without changing the expression's value.

In our exercise, after factoring the common base, we're left with \( (x^{2}+4)^{\frac{3}{2}} (1+(x^{2}+4)^{2}) \). To simplify, we must recognize that the second term inside the parentheses is actually the square of the binomial \(x^{2}+4\). This consistently applied logical chopping helps to make expressions more manageable.

Simplifying expressions is not just about making the algebraic expression shorter in appearance; it's about making it more understandable and easier to evaluate or manipulate in further calculations.

Algebraic Powers

Dealing with algebraic powers requires understanding exponent rules, which are routinely applied in algebra to manipulate expressions involving powers. An algebraic power, like \( a^{n} \), represents the base \( a \) multiplied by itself \( n \) times. The exponent \( n \) denotes how many times you use the base as a factor.

In our original exercise, we encounter powers with fractional exponents. Fractional exponents, such as \( \frac{3}{2} \) and \( \frac{7}{2} \), can be tricky, but they follow the same rules as integer exponents. The numerator represents the power to which the base is raised, while the denominator represents the root. For example, \( a^{\frac{3}{2}} = \sqrt[2]{a^{3}} \) or the square root of \( a^{3} \). Understanding how to manipulate these powers allows for the simplification process, ultimately merging the expression into its sleeker form.