Problem 10 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression. a. \(y^{-3 / 2} y^{5 / 3}\) b. \(x^{-3 / 5} x^{8 / 3}\)

Step-by-Step Solution

Answer

a. \(y^{1/6}\) b. \(x^{31/15}\)

Step 1: Part a: Identifying Exponent Rules (a)

In this part, we will focus on simplifying the expression \(y^{-3/2}y^{5/3}\). To do this, we need to apply the rule:\[(a^m)(a^n)=a^{m+n}\]Since both parts of the expression have the same base of \(y\), we can add the exponents.

Step 2: Part a: Applying Exponent Rules (a)

Using the rule, we can get:\[y^{-3/2}y^{5/3} = y^{-3/2 + 5/3}\]In order to add the exponents, we need to find a common denominator. In this case, the common denominator is 6.

Step 3: Part a: Adding Exponents (a)

Now, we add the exponents with the common denominator:\[y^{-3/2 + 5/3} = y^{-9/6 + 10/6}\]Since the denominators are the same, we can simply add the numerators: \[-9 + 10 = 1\]

Step 4: Part a: Final Simplification (a)

Finally, substitute the result of the addition of the exponents back into the expression:\[y^{-9/6 + 10/6} = y^{1/6}\]So, the simplified expression for part a is \(y^{1/6}\).

Step 5: Part b: Identifying Exponent Rules (b)

In this part, we will focus on simplifying the expression \(x^{-3/5}x^{8/3}\). We will apply the rule:\[(a^m)(a^n)=a^{m+n}\]Since both parts of the expression have the same base of \(x\), we can add the exponents.

Step 6: Part b: Applying Exponent Rules (b)

Using the rule, we can get:\[x^{-3/5}x^{8/3} = x^{-3/5 + 8/3}\]In order to add the exponents, we need to find a common denominator. In this case, the common denominator is 15.

Step 7: Part b: Adding Exponents (b)

Now, we add the exponents with the common denominator:\[x^{-3/5 + 8/3} = x^{-9/15 + 40/15}\]Since the denominators are the same, we can simply add the numerators: \[-9 + 40 = 31\]

Step 8: Part b: Final Simplification (b)

Finally, substitute the result of the addition of the exponents back into the expression:\[x^{-9/15 + 40/15} = x^{31/15}\]So, the simplified expression for part b is \(x^{31/15}\).

Key Concepts

Exponent Rules

Understanding exponent rules is essential when working with algebraic expressions involving powers. In the provided exercise, we explored the fundamental rule \(a^m)(a^n)=a^{m+n}\), which indicates that when multiplying two expressions with the same base, we simply add their exponents. This is because you are essentially multiplying the base by itself for the combined number of times represented by the exponents.

To apply this rule effectively, remember that all terms must share the same base, and that addition is only possible when dealing with exponents, not the bases themselves. When working with variables or unknowns, ensure that they are identical before attempting to simplify through exponent addition. With practice, identifying these opportunities for simplification becomes intuitive, streamlining complex algebraic operations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using various algebraic techniques. In our exercise, we encountered the need for adding fractions as part of the algebraic manipulation process. When the exponents we need to add are in fractional form, as in \(y^{-3/2}y^{5/3}\) or \(x^{-3/5}x^{8/3}\), finding a common denominator is crucial.

This process transforms the exponents into equivalent fractions that can be added together seamlessly. Understanding how to manipulate these expressions algebraically ensures that even complex-looking equations can be broken down into more manageable pieces. Mastering these skills leads to greater confidence in tackling algebraic challenges and solving them with precision.

Mathematical Simplification

The essence of mathematical simplification lies in transforming complex expressions into their simplest form without changing their value. This process makes equations easier to understand and solve. In our exercise, after applying the exponent rules and adding fractions, we simplified the expression to \(y^{1/6}\) or \(x^{31/15}\). Simplification can involve combining like terms, reducing fractions, and eliminating unnecessary complexity.

It's not just about making expressions shorter or more elegant; simplification can reveal structural insights about the mathematical relationships involved, sometimes leading to further revelations about the underlying problem. Practicing simplification helps in developing a more intuitive grasp of algebra and an ability to communicate solutions clearly and concisely.

Fractional Exponents

Fractional exponents, also known as rational exponents, represent a powerful concept in algebra that extends the realm of exponents beyond whole numbers. For instance, in the expressions \(y^{1/6}\) and \(x^{31/15}\), the fractional exponent corresponds to roots. The denominator of a fractional exponent indicates the root—one-sixth power is the same as taking the sixth root of \(y\), and \(31/15\) power is equivalent to the fifteenth root of \(x\) raised to the thirty-first power.

Understanding how to work with fractional exponents is a pivotal step in solving a wide range of algebraic problems. It allows for the manipulation of expressions involving roots in a similar manner to those with whole number exponents, thus broadening the toolkit available for dealing with equations and simplifications in algebra.