Problem 38 - Intermediate Algebra

Question

Simplify each expression. Assume that the variables can be any real number, and use absolute value symbols 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 of a Power Property

We start with the expression \( (243x^{10})^{1/5} \). The power of a power property states that \( (a^m)^n = a^{m imes n} \). Apply this property to the expression: \((243^{1/5}) imes (x^{10 imes (1/5)})\).

Step 2: Simplify the Numerical Base

Evaluate \( 243^{1/5} \). Since 243 is \( 3^5 \), the fifth root of 243 is 3. Therefore, \( 243^{1/5} = 3 \).

Step 3: Simplify the Variable Part

Now simplify \( x^{10 imes (1/5)} \), which is equal to \( x^2 \), because \( 10 imes 1/5 = 2 \).

Step 4: Combine the Simplified Parts

Combine the results from the previous steps: \( 3 \times x^2 \). As the problem indicates to use absolute value symbols and \( x \) is assumed to be any real number, the expression becomes \( 3|x^2| \). Since \( x^2 \) is always non-negative, \(|x^2| = x^2\).

Step 5: Write the Final Answer

Thus, the simplified expression is \( 3x^2 \).

Key Concepts

Power of a Power Property

One of the most important exponent rules is the Power of a Power Property. It helps to simplify expressions involving exponents. This property says:

• If you have a power raised to another power, \((a^m)^n\), you can multiply the exponents. So, \((a^m)^n = a^{m \times n}\).

This rule allows you to transform complicated power expressions into something simpler and more manageable. It’s widely used in algebra to make expressions with exponents easier to handle.
For example, in the expression \((243x^{10})^{1/5}\), apply the property by multiplying the exponents of x. Therefore, \(x^{10} \times (1/5) = x^2\). This step is critical in reducing the complexity of larger exponent expressions.

Fifth Root

Roots are the opposite operation of powers. When we talk about a fifth root, we are asking what number can be multiplied by itself five times to get the original value.

• Think of it like a reverse power; for the fifth root of a number \(a\), written as \(a^{1/5}\), it is the same as solving \(a = b^5\) for \(b\).

This means if \(a = 243\), then \(b = 3\) since \(3^5 = 243\).
Finding the fifth root is a straightforward way to break down numbers into their simpler units, especially when a number is a power of another smaller number.

Absolute Value

The absolute value of a number refers to its distance from zero on a number line. It doesn't care whether the number is positive or negative, only how far it is from zero. Absolute values are shown using two vertical bars, like this: \(|a|\).

• One key property of \(|a|\) is that it's always non-negative.

When applied to expressions, it’s useful to ensure all results are positive. For example, in \(|x^2|\), the value is simply \(x^2\) because squares of real numbers are never negative.
Absolute value is beneficial when simplifying expressions to ensure they represent real scenarios and avoid unwanted negative values.

Real Numbers

Real numbers include all numbers that can be found on the number line. They can be both rational, like \(2\) or \(1/2\), and irrational, like \(\sqrt{2}\) or \(\pi\).

• Rational numbers can be expressed as a fraction of two integers.
• Irrational numbers cannot be expressed as a simple fraction and have non-repeating, infinite decimal parts.

Real numbers are the most extensive set of numbers we use for calculations, and they're crucial in algebra and calculus. When dealing with variables like \(x\) in mathematical expressions, assuming they can be any real number broadens the range of solutions and interpretations.