Problem 81 - Algebra A Combined Function

Question

Simplify, if possible. $$ \left(y^{5}\right)^{4}+\left(y^{2}\right)^{10} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2y^{20} \).

Step 1: Apply the Power of a Power Rule

The power of a power rule states that when you raise an exponent to another exponent, you multiply the exponents. For the term \( (y^5)^4 \), apply this rule to get \( y^{5 \times 4} = y^{20} \). Similarly, for \( (y^2)^{10} \), apply the rule to get \( y^{2 \times 10} = y^{20} \).

Step 2: Combine Like Terms

Both terms \( y^{20} \) and \( y^{20} \) are like terms, meaning they have the same base and exponent. You can combine them by adding the coefficients. The expression \( y^{20} + y^{20} \) simplifies to \( 2y^{20} \) since each term is essentially \( 1y^{20} \).

Key Concepts

Power of a Power Rule

When dealing with exponents, understanding the power of a power rule is crucial. It's all about simplifying expressions where you have an exponent raised to another exponent. Consider the expression \((y^5)^4\). The power of a power rule tells us to multiply the exponents, transforming \((y^5)^4\) into \(y^{20}\). This is because the expression represents the base \(y^{5}\) multiplied by itself four times, essentially leading to a multiplication of the exponents: \(5 \times 4 = 20\).

This rule holds true for any expression of the form \((a^m)^n\), which becomes \(a^{m \times n}\), where \(a\) is the base, and \(m\) and \(n\) are the exponents. Recognizing when and how to apply this rule can make complicated expressions much easier to handle.

Like Terms

Like terms are terms that have the same variable raised to the same power. Knowing how to identify them is a key skill in algebra. In our expression, once we've applied the power of a power rule, we have two terms: \(y^{20}\) and \(y^{20}\). Both these terms have the same variable \(y\) and the same exponent \(20\).

Identifying like terms is essential when you want to simplify expressions, as it allows you to combine them efficiently. In general, when you add or subtract like terms, you operate on the coefficients while keeping the variable part unchanged. For example, adding \(y^{20}\) and \(y^{20}\) yields \(2y^{20}\), because the coefficients \(1\) from each term sum up to \(2\).

• Like terms have identical variables with the same exponents.
• Combine them by adding or subtracting their coefficients.
• This simplifies the expression into something more manageable.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This is where the knowledge of rules like power of a power and the recognition of like terms comes into play. With our expression \(y^{20} + y^{20}\), we saw how applying these rules lead to \(2y^{20}\). Each step simplifies the expression, making it easier to work with.

When simplifying, follow these steps:

• Apply exponent rules to manage powers efficiently, like the power of a power rule.
• Combine like terms by adding their coefficients to keep it simple.
• Check if further simplification is possible, considering other algebraic rules.

Each simplification reduces the complexity of the expression and helps in solving or evaluating it more effortlessly. Becoming proficient in these techniques is like securing your foundational building blocks for more advanced algebraic manipulations.