Problem 25 - Intermediate Algebra

Question

Find each cube root. $$ \sqrt[3]{x^{12}} $$

Step-by-Step Solution

Answer

The cube root of \( x^{12} \) is \( x^4 \).

Step 1: Understand the Problem

To find the cube root of an expression with an exponent, such as \( \sqrt[3]{x^{12}} \), we need to simplify it using the properties of exponents.

Step 2: Use the Exponent Rule for Radicals

The cube root of an expression \( x^n \) can be written using the property: \( \sqrt[3]{x^n} = x^{n/3} \). Apply this rule to the given expression \( \sqrt[3]{x^{12}} \).

Step 3: Calculate the New Exponent

In our expression \( \sqrt[3]{x^{12}} \), the exponent \( 12 \) should be divided by \( 3 \). Therefore, it becomes \( x^{12/3} \).

Step 4: Simplify the Exponent

Perform the division \( 12/3 \) to simplify the exponent. This results in \( x^4 \).

Step 5: Write the Final Answer

After simplifying, we find that the cube root of \( x^{12} \) is \( x^4 \).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial for solving problems like finding the cube root of expressions. Exponents allow us to represent repeated multiplication in a compact form. For instance, \( x^3 \) means \( x \times x \times x \). These properties provide handy shortcuts to simplify expressions:

• Product of Powers: \( x^a \times x^b = x^{a+b} \)


• Power of a Power: \( (x^a)^b = x^{a\times b} \)


• Power of a Product: \( (xy)^a = x^a y^a \)


• Quotient of Powers: \( \frac{x^a}{x^b} = x^{a-b} \)

These principles allow us to manipulate expressions, not just for simplification but also when working with radicals and roots. Remember, an exponent can be a positive integer, zero, or even a fraction, as is the case with radical expressions.

Simplifying Expressions

When dealing with mathematical expressions, especially those with exponents, simplifying them makes them easier to understand and solve. Simplification often involves breaking down complex expressions into simpler or more manageable parts.

• Try to combine like terms. For example, simplify \( x^3 \cdot x^5 \) to \( x^8 \) by using the product of powers property.


• If the expression involves radicals, look for ways to express the radicals using fractional exponents to simplify calculations.

In our original problem, the expression \( \sqrt[3]{x^{12}} \) was simplified by rewriting the cube root as an exponent: \( x^{12/3} \). Performing the division resulted in the simplified expression \( x^4 \). This process always involves following rules while ensuring each step makes the expression clearer.

Exponent Rules for Radicals

Radicals and exponents are tightly linked in mathematics. Understanding the relationship between them is key when performing operations like finding roots. The conversion between radicals and exponents involves fractional exponents.For any mathematical expression, the \( n \)-th root of \( x^a \) can be expressed as \( x^{a/n} \). This is known as the exponent rule for radicals. For example:

• The square root, \( \sqrt{x} \), is expressed as \( x^{1/2} \).


• The cube root, \( \sqrt[3]{x} \), is expressed as \( x^{1/3} \).

In the example \( \sqrt[3]{x^{12}} \), the exponent 12 is divided by 3, reflecting the cube root's nature as a fractional exponent, resulting in \( x^4 \). By utilizing this rule, simplifying expressions involving roots becomes more systematic and predictable, allowing for more straightforward calculations and solutions.