Problem 19 - Basic Engineering Mathematics

Question

Simplify \(\frac{a^{2} b}{a b^{2}-a^{1 / 2} b^{3}}\)

Step-by-Step Solution

Answer

The simplified form is \( \frac{a^{3/2}}{b^2(a^{1/2} - 1)} \).

Step 1: Factor the Denominator

The denominator is \( ab^2 - a^{1/2} b^3 \). We can factor it by extracting the greatest common factor (GCF), which is \( a^{1/2}b^2 \): \[ab^2 - a^{1/2}b^3 = a^{1/2}b^2(ab^{1/2} - b)\].

Step 2: Write the Simplized Expression

Plug the factored denominator into the expression:\[\frac{a^2b}{a^{1/2}b^2(ab^{1/2} - b)}\].

Step 3: Simplify by Canceling Common Terms

In the numerator and denominator, common terms can be simplified. Cancel \(ab^2\) from both:\[\frac{a^{2-1/2}b^{1-2}}{ab^{1/2} - b} = \frac{a^{3/2}b^{-1}}{ab^{1/2} - b}\].Rewrite: \[\frac{a^{3/2}}{b(ab^{1/2} - b)}\].

Step 4: Further Simplification (if possible)

In \( \frac{a^{3/2}}{b(ab^{1/2} - b)} \), check if \(ab^{1/2} - b\) can be factored further. Let us factor \(b\) from the inside:\[\frac{a^{3/2}}{b^2(a^{1/2} - 1)}\].The expression is now fully simplified.

Key Concepts

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler "factors" that, when multiplied together, give back the original polynomial. It’s like reverse distributing.
For the exercise, we had to factor the denominator, which was initially written as \( ab^2 - a^{1/2} b^3 \).
In this situation, factoring involves finding common factors in the terms involved. Understanding that polynomials often share a commonality helps us to start simplifying.

There are several key approaches to factoring polynomials:

• Look for the greatest common factor (GCF), which is a term that is common across all parts of the polynomial.
• Break down the polynomial using algebraic expressions, such as difference of squares or perfect square trinomials.
• Factor by grouping when there are four or more terms.

Factoring helps make expressions easier to manipulate, leading to further simplifications or solutions of equations.

Greatest Common Factor

The greatest common factor (GCF) is a crucial step in simplification and factoring. It’s the largest term that can divide each term of a polynomial evenly.
Finding the GCF helps in reducing expressions, making further algebraic manipulations more straightforward.
In the exercise, to factor \( ab^2 - a^{1/2} b^3 \), we extracted \( a^{1/2}b^2 \) as the GCF.
This involved looking at each term and figuring out what shared components they have in common.
Why GCF is essential:

• Simplifies complex polynomial expressions.
• Helps in revealing hidden roots of polynomial equations.
• Makes fractions with polynomial numerators and denominators easier to work with.

Using the GCF is a vital tool in algebraic simplification, aiding in converting difficult expressions into manageable forms.

Exponents and Radicals

Understanding exponents and radicals is fundamental when simplifying algebraic expressions.Exponents indicate how many times a number, known as the base, is multiplied by itself. For example, in \( a^2 \), \( a \) is the base, and 2 is the exponent, indicating \( a \) is multiplied by itself.
Radicals are the opposite, representing roots like square roots \( (\sqrt{a}) \) or cube roots.
They help in reversing the effect of exponents, particularly when simplifying terms or solving equations.
In the problem, exponents were crucial when reducing terms, such as going from \( a^2 b \) to \( \frac{a^{3/2}}{b^2(a^{1/2}-1)} \) after simplification. This required an understanding of how to apply and manipulate these symbols and their rules, such as the power of a power rule or the product of powers.
Using exponents and radicals appropriately helps to streamline the simplification process, making complex expressions approachable and solvable.