Problem 58 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(\left(x^{3} x^{12}\right)^{1 / 5}\)

Step-by-Step Solution

Answer

The simplified form of \(\left(x^{3} x^{12}\right)^{1 / 5}\) is \(x^{3}\).

Step 1: Use the Product of Powers Property

First, look at the inside of the parenthesis \(x^{3} x^{12}\). By the product of powers property, when multiplying two expressions with the same base, the exponents are added together. Add the exponents 3 and 12 to find the new exponent. This gets you \(x^{3+12} = x^{15}\). Now the expression looks like this: \((x^{15})^{1/5}\).

Step 2: Use the Power of a Power Property

Next, apply the power of a power property to simplify \((x^{15})^{1/5}\). This means that you multiply the exponents 15 and 1/5 together. This gets you \(x^{15*1/5} = x^{3}\).

Step 3: Final Simplified Expression

The mathematical expression is now fully simplified as \(x^{3}\).

Key Concepts

Algebraic Simplification

Algebraic simplification is the process of reducing a complex algebraic expression into its simplest form. It involves using various algebraic rules to condense the expression without changing its value. This process helps make the expression easier to understand and use in further calculations.
Simplification often involves combining like terms, factoring, and applying mathematical properties like the distributive property. When we apply simplification in the exercise \[\left(x^{3} x^{12}\right)^{1/5}\], we aim to make this expression less complex by using known properties of exponents.
It's important in algebra because it allows for simpler calculation and better insight into the behavior of the expression.

• It makes expressions easier to handle and interpret.
• It simplifies calculations and reduces error.
• It aids in finding equivalent expressions.

By simplifying, we turn a potentially confusing expression into one that is more straightforward and understandable.

Product of Powers Property

The Product of Powers Property is a key concept in algebra that helps simplify expressions where you multiply terms with the same base. This rule states that when you multiply two exponents with the same base, you simply add their exponents.
For example, in the term \((x^{3} \cdot x^{12})\), you can simplify it by adding the exponents since the base is the same: \(x^{3+12} = x^{15}\). This is exactly what we did in the first step of the original solution.

• This property is extremely useful for simplifying expressions quickly.
• It works because multiplication of numbers is repeated addition, and exponents express repeated multiplication.
• Remember, it only applies when the base of the terms you are multiplying is identical.

Using the Product of Powers Property reduces complex expressions and is foundational for understanding further properties of exponents.

Power of a Power Property

The Power of a Power Property is another vital tool in exponentiation. This property states that when an exponentiated term is raised to another power, you multiply the exponents. This property helps in reducing nested exponents into a single exponent.
In the example \((x^{15})^{1/5}\), we used the Power of a Power Property by multiplying the exponent 15 by \(1/5\), which gave us \(x^{3}\). This significantly simplifies the expression.

• The property simplifies expressions with multiple layers of exponents.
• It is applicable to any base raised to an exponent and then raised again.
• Makes complex expressions more manageable by reducing double exponents to a single one.

In essence, the Power of a Power Property streamlines the handling of exponents, clarifying and simplifying mathematical expressions.