Problem 53 - A Graphical Approach to College Algebra

Question

Find each product. Assume that all variables represent positive real numbers. $$y^{5 / 8}\left(y^{3 / 8}-10 y^{11 / 8}\right)$$

Step-by-Step Solution

Answer

The product is \( y - 10y^2 \).

Step 1: Distribute the Terms

Apply the distributive property to the expression \[ y^{5/8} (y^{3/8} - 10y^{11/8}) \]Distribute \( y^{5/8} \) to each term inside the parentheses.

Step 2: Multiply Powers of 'y'

Using the product of powers rule \( y^a imes y^b = y^{a+b} \), calculate the product \[ y^{5/8} imes y^{3/8} = y^{(5/8)+(3/8)} = y^{8/8} = y \]

Step 3: Multiply Second Term

Continue multiplying using the product of powers rule: \[ y^{5/8} imes (-10y^{11/8}) = -10y^{(5/8)+(11/8)} = -10y^{16/8} = -10y^2 \]

Step 4: Simplify the Expression

Now that both products are found from Step 2 and Step 3, combine them to simplify the expression:\[ y - 10y^2 \]

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that allows you to break down complex expressions into simpler parts. It essentially states that a term outside the parentheses can be multiplied with each term inside. In our example, the expression can be written as follows:

• Given: \( y^{5/8} (y^{3/8} - 10y^{11/8}) \)
• Apply Distributive Property: \( y^{5/8} \times y^{3/8} - y^{5/8} \times 10y^{11/8} \)

This property helps break down expressions into smaller, manageable pieces. Apply this property first in solving similar problems. Once the terms are separated, we can deal with each multiplication separately. It's important to remember the signs of each term inside the bracket, as they must be carried over during distribution.

Product of Powers Rule

The Product of Powers Rule is used when multiplying expressions with the same base. According to this rule, when you multiply powers, you add their exponents. The formula is given by:

• \( y^a \times y^b = y^{a+b} \)

This rule was applied to multiply the power of\( y \) terms in each part of the expression. Here's how it works step by step:

• First for \( y^{5/8} \times y^{3/8} \), apply the rule: \( y^{5/8 + 3/8} = y^{8/8} = y \).
• Then for \(-10y^{5/8} \times y^{11/8} \), apply it again: \(-10y^{5/8+11/8} = -10y^{16/8} = -10y^2 \).

Remember, always ensure the base remains the same to use this rule effectively. This simplification can greatly reduce the complexity of your algebraic expressions.

Simplifying Expressions

Simplifying Expressions is an essential skill in algebra. After applying properties such as the Distributive Property and Product of Powers, you're often left with an expression that needs to be combined and reduced further. In this example, simplifying resulted in:

• \( y - 10y^2 \)

The key here is to identify like terms and simplify them. In this case, there were no like terms to combine further. Hence, the expression remains \( y - 10y^2 \). Simplification makes your final answer more concise and ready for further uses in mathematics or real-world applications.Always look for opportunities to combine like terms or reduce the expression further. This can involve factoring out common terms or cancelling out terms if they're opposites. Mastering simplification prepares you to tackle more complex algebra problems with efficiency.