Problem 20 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$\left(y^{3}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(y^6\).

Step 1: Identify the base and exponents

In this expression, the base is 'y' and the two exponents are 3 and 2. So we have: \((y^3)^2\).

Step 2: Apply the exponent rule

Now we apply the exponent rule \((a^m)^n = a^{mn}\) to the expression by multiplying the two exponents together: \(y^{(3\times2)}\).

Step 3: Simplify the final expression

After multiplying the exponents, we get the simplified expression: \(y^6\). Since there are no negative exponents in this expression, the answer is already in the desired format. Thus, the simplified expression is \(y^6\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest forms without changing their value. This process makes the expressions easier to work with, especially when solving equations or evaluating functions. In algebra, simplification often includes combining like terms and reducing exponents.
When dealing with exponents, simplification is made easier by following specific rules and guidelines. In our exercise, we were given \((y^3)^2\), which illustrates how an expression can be written in a more straightforward way. Simplification ensures the expression is both efficient and clean, allowing us to focus on solving complex algebraic problems without unnecessary complications.
It's crucial to remember that a simplified expression should not contain any negative exponents. This is an important part of simplifying expressions to maintain the simplicity and readability of the results.

Exponent Rules

Exponent rules are essential tools in algebra that help simplify algebraic expressions involving powers. One of the most commonly used rules is the power of a power rule, which states that when you have an exponent raised to another exponent, you multiply the exponents.
The general form is given by \((a^m)^n = a^{mn}\). This was the rule applied in our exercise, turning \((y^3)^2\) into \(y^{(3\times2)}\) and then to \(y^6\). Understanding and applying these rules allow for efficient manipulation and simplification of expressions.

• **Product of Powers Rule**: When multiplying like bases, add the exponents: \(a^m \times a^n = a^{m+n}\).
• **Quotient of Powers Rule**: When dividing like bases, subtract the exponents: \(a^m / a^n = a^{m-n}\).
• **Negative Exponent Rule**: An expression with a negative exponent can be rewritten as its reciprocal: \(a^{-m} = \frac{1}{a^m}\).

Knowing these exponent rules makes algebra easier and more intuitive.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. It is the foundation of algebra and helps in expressing relationships between quantities. Algebraic expressions can represent real-world scenarios and can be simplified for ease of understanding.
In our example, the expression \((y^3)^2\) was simplified to \(y^6\). This is a classic case of working with exponents within algebraic expressions. By simplifying the expression, we make it more manageable and less cumbersome.
Algebraic expressions often require manipulation to extract meaningful solutions, such as finding values or simplifying models represented by the expressions. Practicing the simplification of expressions with exponents is a crucial step towards mastering more complex algebraic tasks.
Whether in academic problems or real-world applications, the simplification and understanding of algebraic expressions pave the way for logical reasoning and problem-solving.