Problem 14 - Algebra 1

Question

Use the power of a power property to simplify the expression. $$ \left(y^{4}\right)^{5} $$

Step-by-Step Solution

Answer

The simplified expression of \( \left(y^{4}\right)^{5} \) is \( y^{20} \).

Step 1: Identify the base and the exponents

First, identify the base and the exponents. In \( \left(y^{4}\right)^{5} \), y is the base, 4 is the inner exponent, and 5 is the outer exponent.

Step 2: Apply the power of a power property rule

Use the property which states that \( \left(a^{m}\right)^{n}=a^{mn} \). Here, a is y, m is 4, and n is 5.

Step 3: Simplify the expression

Multiply the inner exponent and the outer exponent together: \( y^{4 \times 5}=y^{20} \)

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication in a compact form. They are used in mathematics to denote the number of times a number, known as the base, is multiplied by itself. For example, in the expression \( y^4 \), \( y \) is the base, and 4 is the exponent, which tells us that \( y \) is multiplied by itself 4 times, i.e., \( y \times y \times y \times y \).

Understanding the role of exponents is essential because they simplify complex multiplication problems and make calculations more efficient. Instead of writing out long multiplication sequences, exponents provide a shorthand notation to clearly express these ideas. This concept greatly enhances our capability to handle large numbers or repeated multiplication scenarios in a succinct and manageable format.

Simplifying Expressions Using the Power of a Power Property

Simplifying expressions with exponents involves using specific rules that streamline calculations. One such important rule is the power of a power property.

The power of a power property states that when you have an exponent raised to another exponent, you multiply the exponents together. For instance, in \( (a^m)^n \), the simplification results in \( a^{mn} \). This is because each group of the expression is multiplying the base repeatedly.

• In the given problem: \( \left(y^{4}\right)^{5} \)
• Identify: The base is \( y \), the inner exponent is 4, and the outer exponent is 5.
• Apply: Multiply the exponents: \( 4 \times 5 = 20 \).
• Simplified expression: \( y^{20} \).

This property greatly simplifies the process, reducing the need for lengthy calculations, making expressions easier to handle.

Algebraic Expressions and Their Components

An algebraic expression is a combination of numbers, variables (like \( y \)), and operators (such as addition, subtraction, multiplication, and division) that represent a mathematical relationship. Variables are often used in algebra to represent unknown values or to generalize calculations.

In the expression \( \left(y^4\right)^5 \), \( y \) is a variable. Variables can change, and are often used in algebra to form equations or expressions that model real-world scenarios. The goal with algebraic expressions is often to simplify them, making them easier to work with or solve when paired with equations.

Understanding how to manipulate expressions, including applying exponent rules, such as the power of a power property, is crucial in algebra. It allows us to understand relationships and solve problems effectively, providing a foundation for more advanced mathematical concepts.