Problem 90 - Elementary and Intermediate Algebra

Question

Perform the multiplications. All variables represent positive real numbers. See Example 9. $$ y^{2 / 5}\left(y^{-2 / 5}+y^{3 / 5}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( 1 + y \).

Step 1: Distribute the Term

The expression given is \( y^{2/5} (y^{-2/5} + y^{3/5}) \). Using the distributive property, this can be expanded into two separate products: \( y^{2/5} \times y^{-2/5} + y^{2/5} \times y^{3/5} \).

Step 2: Simplify Each Term

For the first term, use the property of exponents \( a^m \times a^n = a^{m+n} \). Thus, \( y^{2/5} \times y^{-2/5} = y^{(2/5) + (-2/5)} = y^{0} \). For the second term, \( y^{2/5} \times y^{3/5} = y^{(2/5) + (3/5)} = y^{5/5} = y^1 \).

Step 3: Substitute and Simplify

Substitute the simplified terms back into the expression: \( y^0 + y^1 \). Any number raised to the power of zero is 1, so \( y^0 = 1 \). Thus, the expression simplifies to \( 1 + y \).

Key Concepts

Distributive Property

The distributive property is a foundational tool in algebra, allowing a multiplication operation to be distributed across terms inside parentheses. This property states that for any numbers or variables,

• \( a(b + c) = ab + ac \)

In our specific example, the expression \(y^{2/5}(y^{-2/5} + y^{3/5})\) can be distributed as \(y^{2/5} \times y^{-2/5} + y^{2/5} \times y^{3/5}\).
This step is essential as it lays the groundwork to simplify complex expressions by breaking them into more manageable parts.

Properties of Exponents

Exponents have specific properties that make simplifying expressions easier. Understanding these rules is crucial when dealing with exponential terms, such as those found in this exercise.
Here are some key properties:

• Product of Powers: \( a^m \times a^n = a^{m+n} \). This allows you to add the exponents when multiplying with the same base.
• Zero Exponent: Any non-zero number raised to the power of zero is 1, i.e., \( a^0 = 1 \).

Using these properties, we simplify expressions like \( y^{2/5} \times y^{-2/5} \) to \( y^{0} = 1 \) and \( y^{2/5} \times y^{3/5} \) to \( y^1 \).
These properties are vital for efficient computation and simplification in algebra.

Simplifying Expressions

Simplifying expressions is the process of making a complex equation easier to understand and solve. In this problem, we've started by using the distributive property, and then applied the properties of exponents. The final step involves substituting simplified terms back into the expression.
Starting with \( y^{0} + y^{1} \), we know from the zero exponent property that \( y^{0} = 1 \). Thus, the simplified form becomes \( 1 + y \).
This conversion to a simpler form is critical for making calculations quicker and avoiding potential mistakes while working with more intricate algebraic equations.