Problem 36 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ \left(y^{7}\right)^{5} $$

Step-by-Step Solution

Answer

The expression \((y^7)^5\) simplifies to \(y^{35}\).

Step 1: Identify the Outer Exponentiation

The expression \((y^7)^5\) involves raising \(y^7\) to the power of 5. Here, \(y^7\) is the base and 5 is the outer exponent.

Step 2: Apply the Power Rule

The power rule for exponents states that \((a^m)^n = a^{m \cdot n}\). In our problem, this translates to \((y^7)^5 = y^{7 \cdot 5}\).

Step 3: Simplify the Exponent

Calculate the product of the exponents: \(7 \times 5 = 35\). Therefore, \((y^7)^5 = y^{35}\).

Step 4: Write the Final Expression

After simplifying, the expression \((y^7)^5\) simplifies to \(y^{35}\).

Key Concepts

Power Rule

The power rule in exponentiation is a fundamental principle that allows you to simplify expressions involving exponents. When you have an expression of the form \((a^m)^n\), the power rule states that this can be simplified to \(a^{m \cdot n}\). This means you multiply the exponents together instead of adding them. This rule is especially helpful when dealing with nested exponents or when simplifying complex expressions. Some key points to remember about the power rule:

• It applies to powers of powers, simplifying them into a single exponent.
• The rule helps to reduce multi-layered algebraic expressions to simpler forms.
• It is universally applicable to any base, whether a variable or a number.

The power rule is not only about mechanics but about understanding the multiplication of repeated operations. In our example, \((y^7)^5\), applying this rule means that multiplying 7 and 5 gives \(y^{35}\), streamlining the expression efficiently.

Exponents

Exponents are the mathematical shorthand for repeated multiplication. They describe how many times a number, known as the base, is to be multiplied by itself. For instance, in the expression \(y^7\), the base \(y\) is raised to the 7th power, meaning \(y\) is multiplied by itself seven times. Exponents are essential for expressing large numbers in a concise way. Understanding exponents involves recognizing them as more than just numbers or superscripts:

• Exponents simplify writing and calculations; higher exponents mean bigger numbers quickly.
• The base raised to an exponent yields a power, which captures the repeated multiplicative action.
• Combining exponents involves specific rules, such as the power rule, that can streamline computations.

Exponents appear in various forms across different expressions, playing crucial roles from basic algebra to advanced calculus. They help clarify not only the scale of numbers but also provide insights into the nature of growth and decay in different scientific contexts.

Simplifying Expressions

Simplifying expressions is a process of rewriting them in an easier and more understandable form. Simplification often makes use of exponent rules, such as the power rule, to combine and reduce terms to their simplest form. In our example, \((y^7)^5\), simplifying involves applying the power rule to condense the expression into \(y^{35}\). Here's how simplifying works with exponents:

• Identify opportunities to apply rules that will condense terms.
• By reducing complexity, you make the expressions easier to work with in solving equations or problems.
• Simplification often reveals underlying relationships within mathematical problems.

Simplifying expressions is not just about making them shorter, but also about making them more comprehensible. In mathematics, clarity is key, and simplification serves to eliminate unnecessary complication, providing a clear view of the relationships among variables and constants.