Problem 599 - Elementary Algebra

Question

In the following exercises, simplify. $$ \left(u^{12} v^{18}\right)^{\frac{1}{6}} $$

Step-by-Step Solution

Answer

The simplified form is \( u^2 v^3 \).

Step 1: Apply the Power Rule

The Power Rule states that \[ (a^m)^n = a^{m \times n} \] Apply this rule to \[(u^{12} v^{18})^{\frac{1}{6}} \]

Step 2: Distribute the Exponent

Distribute the \(\frac{1}{6} \) exponent to both \(u^{12} \) and \(v^{18} \). This results in: \[ (u^{12})^{\frac{1}{6}} (v^{18})^{\frac{1}{6}} \]

Step 3: Simplify Each Term

Simplify \(u^{12 \times \frac{1}{6}} \) and \(v^{18 \times \frac{1}{6}} \). \[ u^{12 \times \frac{1}{6}} = u^2 \] \[ v^{18 \times \frac{1}{6}} = v^3 \]

Step 4: Combine Simplified Terms

Combine the simplified terms: \[ u^2 \times v^3 = u^2 v^3 \]

Key Concepts

Power Rule

The Power Rule is a fundamental principle in algebra when dealing with exponents. It states that for any base a raised to an exponent m, and this result raised to another exponent n, the result is the base raised to the product of the two exponents. Mathematically, this can be expressed as \( (a^m)^n = a^{m \times n} \). This rule helps to simplify expressions where a power is raised to another power. In our example, we start with the expression \[ (u^{12} v^{18})^{\frac{1}{6}} \]. Applying the Power Rule, we use \[ (a^m)^n = a^{m \times n} \] individually to both parts: \[ (u^{12})^{\frac{1}{6}} \] and \[ (v^{18})^{\frac{1}{6}} \].

Distributing Exponents

Distributing exponents is the process where an external exponent is applied to each component inside a product. When an expression like \((ab)^c\) is given, you can distribute the exponent c to each base inside the parentheses: \((a^b)^c = a^{bc} \). In our case, we have the expression \[ (u^{12} v^{18})^{\frac{1}{6}} \]. By distributing the \({\frac{1}{6}}\) exponent, we separately apply it to both \(u^{12}\) and \(v^{18}\). This gives us \[ (u^{12})^{\frac{1}{6}} (v^{18})^{\frac{1}{6}} \]. Now, each part can be simplified further using previously learned rules.

Simplifying Exponents

Simplifying exponents involves reducing the expression to its simplest form. Here, we address each term separately. For \((u^{12})^{\frac{1}{6}}\), we use the Power Rule: \[ (u^{12})^{\frac{1}{6}} = u^{12 \times \frac{1}{6}} = u^2 \]. Similarly, for \((v^{18})^{\frac{1}{6}}\), applying the same rule: \[ v^{18 \times \frac{1}{6}} = v^3 \]. Putting these together, we get \((u^{12})^{\frac{1}{6}} (v^{18})^{\frac{1}{6}} = u^2 v^3 \). Each term is now simplified to a much easier and more manageable form.

Exponential Expressions

Exponential expressions are mathematical phrases that involve exponents. They often require careful manipulation using rules like the power rule and distributing exponents to simplify. When dealing with exponential expressions:

• Identify the base and the exponents.
• Apply relevant rules like the power rule to make calculations simpler.
• Simplify each part before combining them.

In our exercise, the initial exponential expression was \[ (u^{12} v^{18})^{\frac{1}{6}} \]. By using the Power Rule and distributing exponents, we simplified it step-by-step to get the result: \[ u^2 v^3 \]. These systematic steps make working with exponential expressions more efficient and less intimidating.