Problem 22 - College Algebra with Modeling and Visualization

Question

Find the cube root of the number. $$ 8 x^{6} $$

Step-by-Step Solution

Answer

The cube root of \(8x^6\) is \(2x^2\).

Step 1: Identify the Given Expression

The expression we need to find the cube root of is \(8x^6\).

Step 2: Break Down the Expression

Notice that the expression \(8x^6\) can be broken down into a numerical part \(8\) and a variable part \(x^6\).

Step 3: Find the Cube Root of the Numerical Part

The cube root of \(8\) is \(2\) because \(2^3 = 8\).

Step 4: Find the Cube Root of the Variable Part

For the variable \(x^6\), use the rule that says the cube root of \(x^a\) is \(x^{a/3}\). Thus, the cube root of \(x^6\) is \(x^{6/3} = x^{2}\).

Step 5: Combine the Results

Combine the results from Step 3 and Step 4. The cube root of \(8x^6\) is \(2x^{2}\).

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical phrases that can include numbers, variables, and exponents, often arranged in a sum or difference. In simple terms, they are expressions made up of terms. Each term in a polynomial can have a coefficient (a number), a variable raised to a specific power (exponent), or both.

Key features of polynomial expressions include:

• They consist of one or more terms added or subtracted together.
• Each term has a constant multiplier known as the coefficient.
• Variables within terms might have exponents, typically whole numbers.

For example, in the polynomial expression \(8x^6\), \(8\) is the coefficient and \(x^6\) signifies that the variable \(x\) is raised to the sixth power. Understanding how to work with these expressions is crucial when dealing with operations such as finding roots, like cube roots.

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. In mathematical notation, an exponent is usually written as a superscript number next to the base number or variable. This indicates how many times the base is multiplied by itself.

For instance, in the term \(x^6\), the base is \(x\) and the exponent is \(6\), meaning \(x\) is multiplied by itself six times: \(x \times x \times x \times x \times x \times x\).

Exponents play a significant role in finding roots, such as cube roots. A cube root of a number reverses the action of cubing it – raising it to the power of three. For variables with exponents, the rule for finding the cube root is to divide the exponent by three, as seen in the operation where \(x^6\) becomes \(x^{6/3} = x^2\). This understanding helps simplify expressions involving exponents effectively.

Algebraic Operations

Algebraic operations are mathematical procedures that involve manipulating algebraic expressions through addition, subtraction, multiplication, division, and taking roots. These operations allow us to simplify, solve, and transform expressions to express relationships or find values.

In the context of cube roots, the operation involves determining a value which, when cubed, gives the original expression. The cubic root process can be broken into steps involving separate handling of numerical and variable components.

To find the cube root of \(8x^6\):

• The numerical part \(8\) simplifies to \(2\), since \(2^3 = 8\).
• The variable part \(x^6\) simplifies to \(x^{2}\), because of the operation performing \(6/3\) on the exponent.

These steps show how breaking down a complex algebraic expression using algebraic operations can yield simpler, more manageable expressions. Understanding these operations is key to mastering algebra.