Problem 66 - Elementary and Intermediate Algebra

Question

Use the product and power rules for exponents to simplify each expression. $$ \left(v^{5}\right)^{2}\left(v^{3}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \(v^{22}\).

Step 1: Identify the Expression

We are given the expression \((v^5)^2 (v^3)^4\) to simplify using the product and power rules of exponents.

Step 2: Apply the Power Rule

The power rule states that \((a^m)^n = a^{m \cdot n}\). Apply this to \((v^5)^2\) and \((v^3)^4\).\((v^5)^2 = v^{5 \times 2} = v^{10}\)\((v^3)^4 = v^{3 \times 4} = v^{12}\)

Step 3: Apply the Product Rule

The product rule states that \(a^m \cdot a^n = a^{m+n}\). Apply this to \(v^{10} \cdot v^{12}\).\(v^{10} \cdot v^{12} = v^{10 + 12} = v^{22}\)

Step 4: Combine the Steps

Combine the results from Step 2 and Step 3. The simplified expression results from multiplying the products obtained:\((v^5)^2 (v^3)^4 = v^{22}\).

Key Concepts

Product Rule

The product rule is a fundamental concept in exponentiation that helps simplify expressions where the same base is multiplied. If you ever encounter something like \(a^m \cdot a^n\), the product rule allows you to simplify it to \(a^{m+n}\). This combines the exponents by adding them together because multiplying powers with the same base means the powers are essentially adding more of that base.For example, if you have \( v^{10} \cdot v^{12} \), you apply the product rule. The base \(v\) stays the same, and the exponents 10 and 12 are simply added together, resulting in \(v^{22}\). This rule makes dealing with powers much easier and helps in solving many algebraic expressions efficiently.

Power Rule

The power rule is used when raising an exponent to another power, such as \((a^m)^n\). In this rule, the exponents are multiplied together: \((a^m)^n = a^{m \cdot n}\). This is useful when you need to simplify an expression where an exponent is already applied to a power.Consider applying the power rule to the expression \((v^5)^2\). You multiply the exponent 5 by 2, which equals 10, so you get \(v^{10}\). Similarly, for \((v^3)^4\), you multiply 3 by 4 to get 12, resulting in \(v^{12}\). The power rule saves time and ensures accuracy when dealing with multiple layers of exponents.

Simplifying Expressions

Simplifying expressions is about transforming a complex expression into its simplest form. This process often involves using exponent rules like the product and power rules. When you have a complicated expression, breaking it down using these rules can make it much more manageable. For our expression \((v^5)^2 (v^3)^4\), you first apply the power rule to each separate part, getting \(v^{10}\) and \(v^{12}\). Then, use the product rule to combine these results. By adding the exponents 10 and 12, the expression becomes \(v^{22}\). This single final power of \(v\) is the simplified form of the original expression, demonstrating the efficiency of consistently applying these rules. It's like untangling a knot in algebra, leaving you with a clean, straightforward result.