Problem 71 - Applied Calculus

Question

71-86. Simplify. $$ \left(x^{3} \cdot x^{2}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(x^{10}\).

Step 1: Apply the Product of Powers Rule

The expression inside the parentheses is \(x^3 \cdot x^2\). According to the product of powers rule, which states \(a^m \cdot a^n = a^{m+n}\), you can combine the exponents. Therefore, \(x^3 \cdot x^2\) simplifies to \(x^{3+2}=x^5\).

Step 2: Apply the Power of a Power Rule

Next, you have the expression \((x^5)^2\). Use the power of a power rule, which states \((a^m)^n = a^{m \cdot n}\), to simplify. Here, \((x^5)^2 = x^{5 \cdot 2} = x^{10}\).

Key Concepts

Product of Powers Rule

Understanding the **Product of Powers Rule** is fundamental in working with exponents. This rule tells us a simple way to combine powers that have the same base. If you have two exponential terms with the same base, like \(a^m\) and \(a^n\), the rule \(a^m \cdot a^n = a^{m+n}\) allows you to add their exponents.
This is because multiplying the same base together essentially increases the number of times we multiply the base by itself. For example, in the expression \(x^3 \cdot x^2\), \(x^3\) means \(x\) is multiplied by itself 3 times and \(x^2\) means multiplying \(x\) by itself 2 more times. Altogether, you multiply \(x\) by itself 5 times, which results in \(x^5\).
The product of powers rule makes simplifying expressions easier and helps in breaking down more complex problems into manageable parts. By combining like terms, everything with the same base becomes clearer and is easier to handle.

Power of a Power Rule

The **Power of a Power Rule** allows you to simplify expressions where the same base is raised to multiple powers. This rule simplifies the process of working with exponents. The formula \((a^m)^n = a^{m \cdot n}\) shows us that when you raise a power to another power, you multiply the exponents together.
Let's apply it to \((x^5)^2\). Here, \(x^5\) is our base raised to the power of 2. According to the rule, you multiply the exponents: \(5 \cdot 2\). The result simplifies to \(x^{10}\).
This rule helps us work through nested exponents efficiently and can be crucial when handling more complex mathematical expressions. By understanding and applying these exponent rules, larger expressions become less intimidating, and solving them becomes straightforward.

Simplification of Expressions

Simplifying expressions involving exponents can significantly reduce the complexity of algebraic problems. It involves using rules from exponentiation to combine, reduce, or rewrite expressions in a simpler form. In using the **Product of Powers Rule** and **Power of a Power Rule**, expressions that might initially seem complicated can be broken down into more manageable parts.
Take the example \(\left(x^{3} \cdot x^{2}\right)^{2}\). By systematically using these rules, we simplified the expression first to \(x^5\) and then to \(x^{10}\). Each step applies a specific rule that either combines exponents by addition or modifies them by multiplication.
Simplification not only aids in easier computation but also ensures clarity and efficiency in problem-solving. It allows mathematicians to quickly assess, compute, and verify outcomes. Mastering these simplification techniques provides a strong foundation for more advanced studies in algebra and calculus.