Problem 138 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. $$ \left(\frac{2 a}{b^{3}}\right)^{4} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression with only positive exponents and simplify: $\left(\frac{2^4\times a^{4}}{b^{3\times4}}\right)$ Answer: $\frac{16a^4}{b^{12}}$

Step 1: Rewrite the expression using exponent properties

Using the given expression, we will first apply the power to the terms inside the parentheses. $$ \left(\frac{2^4\times a^{4}}{b^{3\times4}}\right) $$

Step 2: Simplify the expression

Now, we'll simplify the expression by replacing the terms with their computed values. $$ \frac{16\times a^4}{b^{12}} $$

Step 3: Final expression

The final expression with only positive exponents is: $$ \frac{16a^4}{b^{12}} $$

Key Concepts

Exponent Properties

Understanding how to work with exponents is crucial for tackling a range of math problems, especially when simplifying algebraic expressions. When an entire fraction is raised to a power, as in the provided exercise,
you need to apply the exponent to both the numerator and the denominator. This stems from one of the key exponent properties, which is that (a/b)^n = a^n / b^n. Breaking down this rule step by step helps clarify its application.

• Start by taking each part of the fraction and raising it to the given power.
• If the base is a product (like 2a), that entire product is raised to the power, which means applying the power to each factor individually (2⁴ and a⁴, respectively).
• On the denominator's side, the power outside the parentheses multiplies the current exponent of the base (3⁴).

This leads to each component within the initial fraction being affected by the power outside, creating a new expression where all bases now have positive exponents, complying with the exercise's requirements.

Simplifying Expressions

Once you've applied the exponent properties, the next step is to simplify the expression. This means calculating the values of the powers you've established and condensing the expression down to its simplest form.
For example, in the given exercise, once we've determined that we need to calculate 2⁴ and b^3*4, the simplification process involves seeing these as basic calculations: 2⁴ equals 16, and b¹² remains as the exponent indicates a single base to a power.
Therefore, 16a⁴ over b¹² is the simplified expression.
This simplification strategy is widely applicable in algebra, allowing you to transform complex expressions into more manageable forms. Remember while simplifying, keep in mind any potential for further simplification, such as factoring, reducing algebraic fractions, or canceling common terms, though in this exercise, those steps were not necessary.

Algebraic Fractions

Algebraic fractions are simply fractions that contain algebraic expressions in the numerator, the denominator, or both. Handling them can be intimidating at first, but by understanding a few rules, you can easily manage these types of problems.
In the context of the exercise, we're not only dealing with an algebraic fraction (2a/b³) but also with the application of an exponent to that fraction, (2a/b³)⁴.
Here are some critical points to keep in mind when working with algebraic fractions:

• Like numerical fractions, the goal is to simplify them as much as possible.
• Keeping exponents positive ensures the fraction stays in its simplified form, consistent with mathematical convention.
• Do not overlook applying the power to both the numerator and the denominator, which allows the fraction to maintain its equivalence.
• Always simplify within the fraction before applying powers; this prevents unnecessary complexity.

By following these points, you'll improve your ability to tackle algebraic fractions and efficiently work through various mathematical expressions.