Problem 65 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Apply Product of Powers Rule

First, simplify the expression within the parentheses using the product of powers rule. This rule states that when you multiply two exponents with the same base, you add the exponents: \(x^a \cdot x^b = x^{a+b}\).Thus, \(x^3 \cdot x^2 = x^{3+2} = x^5\).The expression becomes:\[(x^5)^2.\]

Step 2: Apply Power of a Power Rule

Now, simplify \((x^5)^2\) using the power of a power rule, which states that when raising a power to another power, you multiply the exponents:\((x^m)^n = x^{m \cdot n}\).So, \((x^5)^2 = x^{5 \cdot 2} = x^{10}.\)The simplified expression is:\[x^{10}.\]

Key Concepts

Product of Powers Rule

The Product of Powers Rule is a fundamental concept when dealing with exponents in algebra. It provides a straightforward way to simplify expressions where you multiply terms with the same base. When two exponents share this common base, you simply add the exponents. For example, if you have \( x^a \) multiplied by \( x^b \), the result is \( x^{a+b} \). This is because the base remains the same, and you are essentially counting how many times the base is used as a factor in total.

• Example: \( x^3 \cdot x^2 \) can be rewritten as \( x^{3+2} = x^5 \).

The simplicity of this rule makes it very useful, especially when you encounter long strings of the same base. It not only streamlines computations but also helps to recognize patterns in algebraic expressions for further simplification.

Power of a Power Rule

The Power of a Power Rule involves expressions where an exponent is raised to another exponent. This rule is useful when simplifying expressions like \((x^m)^n\). Applying this rule involves multiplying the two exponents instead of adding them. Thus, it converts the expression into \(x^{m \cdot n}\).

• Example: \((x^5)^2\) simplifies to \(x^{5 \times 2} = x^{10}\).

This formula comes in handy when dealing with nested exponentiations, ensuring that all operations are efficiently simplified. Understanding this helps to grasp how nested powers condense into a single exponentiated term.

Simplification of Expressions

Simplifying expressions is the process of reducing an expression into the simplest form possible, making it easier to understand or to use in further calculations. This often involves applying rules such as the Product of Powers and Power of a Power rules. By using these techniques, you can transform complex expressions into more manageable forms.

• Start by looking for common bases and apply the Product of Powers Rule when multiplying terms.
• Use the Power of a Power Rule when exponents are raised to another power to minimize the expression.

In our original problem, starting from \((x^3 \cdot x^2)^2\), we simplified it to \(x^{10}\) by using these rules in sequence. Always aim to rewrite algebraic expressions in their simplest form to enhance clarity and mathematical insight.