Problem 38 - Elementary and Intermediate Algebra

Question

Simplify each expression. Assume that the variables can be any real number, and use absolute value symbols when necessary. See Example 2 . $$ \left(243 x^{10}\right)^{1 / 5} $$

Step-by-Step Solution

Answer

The simplified expression is \(3x^2\).

Step 1: Apply the Power Rule

First, use the power of a power rule: \[(a^m)^n = a^{m \cdot n}\]For our expression \((243x^{10})^{1/5}\), apply the rule separately to each part, giving:\[243^{1/5} \cdot (x^{10})^{1/5}\].

Step 2: Simplify the Numerical Base

Next, take the fifth root of 243. We know that \(243 = 3^5\), so \(243^{1/5} = (3^5)^{1/5} = 3\). Thus,\(243^{1/5} = 3\).

Step 3: Simplify the Variable Expression

Consider \((x^{10})^{1/5}\). Use the power of a power rule again,\((x^{10})^{1/5} = x^{10 \cdot \frac{1}{5}} = x^2\).

Step 4: Combine Results

Combine the results from Step 2 and Step 3:\[3 \cdot x^2 = 3x^2\].

Key Concepts

The Power Rule

It's essential to understand the concept of the power rule when dealing with exponents. The power rule helps you simplify expressions with exponents effectively. The rule states:

• For any base \(a\) and exponents \(m\) and \(n\), \[(a^m)^n = a^{m \cdot n} \]

This means you multiply the exponents together when raising a power to another power. This method can simplify expressions significantly, particularly when dealing with multiple layers of exponents.

In our original problem, \[(243x^{10})^{1/5} \], applying the power rule separately to each part allowed us to transform the expression into \[243^{1/5} \cdot (x^{10})^{1/5} \]. It's helpful to remember that this rule applies individually to each component within the parentheses. Simple yet powerful, this rule is fundamental for simplifying complex exponential expressions.

Simplifying Expressions

Simplifying expressions is about reducing them to their most basic form. By doing this, it makes further calculations easier and more understandable. Simplification often involves a mix of arithmetic operations and applying specific algebraic rules, like the power rule.

• Breaking down components: Address each part of the expression individually, such as constants, variables, and exponents.
• Applying algebraic principles: Use rules like the power rule, combining like terms, or factoring.
• Looking for patterns: Recognize repeated bases or terms that allow for simplification.

In our context, after applying the power rule, \[243^{1/5} \] and \[(x^{10})^{1/5} \] become prime candidates for further simplification. Once simplified, they turn into more straightforward components: \[3 \] and \[x^2 \], respectively. Realize that simplifying expressions isn't just about following steps but also understanding the structure of the problem to make intelligent decisions as you process each part.

Roots of Numbers

Understanding the roots of numbers is crucial in many algebraic processes, such as simplifying expressions. The "root" of a number is another number that, when multiplied by itself a certain number of times, gives the original number. For instance, the fifth root of a number \(a\) is \(a^{1/5}\).

• Breaking down numbers to their base: Recognize if a number is a power of another. It helps in finding roots quickly.
• Applying root calculation: Use the knowledge of exponents and multiplication to find roots.

For example, to find \[243^{1/5} \], identify that \[243 = 3^5 \]. This realization simplifies the calculation to \[(3^5)^{1/5} = 3\].

Grasping roots is fundamental because it reduces larger numbers to more manageable parts, enabling simpler arithmetic and algebraic manipulations in expressions.