Problem 81
Question
Rationalize the denominator of each radical expression. Assume that all variables represent nonnegative real numbers and that no denominators are \(0 .\) $$\frac{6}{\sqrt[3]{x^{2}}}$$
Step-by-Step Solution
Verified Answer
The rationalized form is \( \frac{6\sqrt[3]{x}}{x} \).
1Step 1: Identify the Goal
The goal is to rationalize the denominator of the expression \( \frac{6}{\sqrt[3]{x^{2}}} \). This means we need to eliminate the cube root from the denominator.
2Step 2: Multiply by the Conjugate Factor
To eliminate the cube root, multiply both the numerator and the denominator by a factor that will make the denominator a complete cube. In this case, multiply by \( \sqrt[3]{x} \) to make the denominator \( x \). So, multiply by \( \frac{\sqrt[3]{x}}{\sqrt[3]{x}} \).
3Step 3: Simplify the Denominator
After multiplying, the denominator becomes \( \sqrt[3]{x^{2}} \times \sqrt[3]{x} = \sqrt[3]{x^{3}} = x \) since \( \sqrt[3]{x^{3}} \) is simply \( x \).
4Step 4: Simplify the Numerator
The numerator becomes \( 6 \times \sqrt[3]{x} = 6\sqrt[3]{x} \).
5Step 5: Write the Final Expression
Combine the simplified numerator and denominator into the fraction \( \frac{6\sqrt[3]{x}}{x} \).
6Step 6: Check for Simplifications
Since \( x \) is in the denominator and there are no common factors in the numerator and denominator, the expression is in its simplest form.
Key Concepts
Cube RootsSimplifying FractionsAlgebraic Expressions
Cube Roots
A cube root is a special type of root that deals with three identical factors. In mathematical terms, it is a number that, when multiplied by itself twice, gives the original number.
To denote this, we use the symbol \( \sqrt[3]{\ } \). For example, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \). Finding the cube root of an expression might seem tricky, but once you strip it down to basics, it's quite manageable.
When working with algebraic expressions, identifying cube roots involves analyzing powers and ensuring they form complete cubes. For instance, in \( \sqrt[3]{x^2} \), this represents a cube root where \( x^2 \) is not a perfect cube, meaning adjustments are necessary to simplify expressions fully.
This is often done by rationalizing the denominator, that is, by multiplying with an appropriate factor, we make sure the expression under the root becomes a perfect cube.
To denote this, we use the symbol \( \sqrt[3]{\ } \). For example, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \). Finding the cube root of an expression might seem tricky, but once you strip it down to basics, it's quite manageable.
When working with algebraic expressions, identifying cube roots involves analyzing powers and ensuring they form complete cubes. For instance, in \( \sqrt[3]{x^2} \), this represents a cube root where \( x^2 \) is not a perfect cube, meaning adjustments are necessary to simplify expressions fully.
This is often done by rationalizing the denominator, that is, by multiplying with an appropriate factor, we make sure the expression under the root becomes a perfect cube.
Simplifying Fractions
At times, fractions can seem complex, especially when radicals or roots are involved. Simplifying fractions means we want to reduce the fraction to its simplest form.
Let's take a practical approach to understand this. Consider the fraction \( \frac{6}{\sqrt[3]{x^2}} \). In this case, simplifying involves eliminating the cube root from the denominator. This process makes the fraction more manageable and easier to work with.
Let's take a practical approach to understand this. Consider the fraction \( \frac{6}{\sqrt[3]{x^2}} \). In this case, simplifying involves eliminating the cube root from the denominator. This process makes the fraction more manageable and easier to work with.
- Identify any radicals in the numerator or denominator.
- Apply the appropriate factor to remove the radical (here, \( \sqrt[3]{x} \)).
- Check for any common factors that can be canceled out.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. They can sound daunting at first, but let's break it down.
In its essence, an algebraic expression like \( \frac{6}{\sqrt[3]{x^2}} \) is trying to convey a relationship or a calculation in which the exact value of \( x \) isn't known.
To rationalize this algebraic expression, we strategically eliminate the cube root to simplify the entire expression. This involves balancing the numerator and the denominator, making sure our new expression \( \frac{6\sqrt[3]{x}}{x} \) accurately reflects the same value as the original _but in a more friendly guise._ Remember:
In its essence, an algebraic expression like \( \frac{6}{\sqrt[3]{x^2}} \) is trying to convey a relationship or a calculation in which the exact value of \( x \) isn't known.
To rationalize this algebraic expression, we strategically eliminate the cube root to simplify the entire expression. This involves balancing the numerator and the denominator, making sure our new expression \( \frac{6\sqrt[3]{x}}{x} \) accurately reflects the same value as the original _but in a more friendly guise._ Remember:
- Elementary operations, such as multiplication and division, allow us to transform these expressions.
- Look out for operations that may help simplify the expression or make it easier to interpret.
Other exercises in this chapter
Problem 80
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