Problem 49 - Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \frac{\sqrt[3]{48 x^{7}}}{\sqrt[3]{6 x}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2x^2\).

Step 1: Recognize Cube Root Property

Use the property that \[ \frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}} \] to combine the expression under a single cube root. Apply this property to simplify the expression to \[ \sqrt[3]{\frac{48x^{7}}{6x}}. \]

Step 2: Simplify the Quotient

Simplify the fraction inside the cube root by dividing the coefficients and subtracting the exponents of like bases: \[ \frac{48x^{7}}{6x} = \frac{48}{6} \times \frac{x^{7}}{x} = 8x^{6}. \] Now the expression becomes \[ \sqrt[3]{8x^{6}}. \]

Step 3: Simplify the Cube Root

Recognize that \(8 = 2^3\), so \(\sqrt[3]{8} = 2\). For \(x^6\), since \(\sqrt[3]{x^6} = x^{6/3} = x^2\), apply the cube root to each term: \[ \sqrt[3]{8x^{6}} = 2x^2. \]

Key Concepts

Cube Root

The cube root is essentially the operation that reverses raising a number to the third power. While taking a square root answers the question, "What number times itself gives me this number?", a cube root answers, "What number times itself three times gives me this number?" This is denoted as \( \sqrt[3]{...} \). Understanding cube roots is essential when working with expressions under the cube root sign, as it allows you to simplify such terms further.
For example, if you have \( \sqrt[3]{8} \), you're looking for a number that, when multiplied by itself three times, results in 8. Since \( 2 \times 2 \times 2 = 8 \), the answer is 2.
Cube roots also play a crucial role in simplifying expressions. Simplifying complex expressions often requires recognizing parts of the number that are perfect cubes (like 8 in the example) so that they can be 'brought out' of the root.

Quotient Rule for Exponents

The quotient rule for exponents helps you simplify expressions that contain division and variables raised to powers. This rule states that when dividing like bases, you subtract the exponents:

• For example, \( \frac{x^a}{x^b} = x^{a-b} \).

This rule applies when the bases are the same, and it can simplify expressions significantly when used correctly.
In the context of the provided exercise, this rule helps reduce the fraction \( \frac{x^7}{x} \). By applying the rule, you subtract the exponent in the denominator from the exponent in the numerator:
\[ x^7 \div x = x^{7-1} = x^6. \]
This simplification is crucial in reducing complex expressions into a simpler cube root form that can be further managed. By learning this rule, you gain an important tool for tackling a range of algebraic problems.

Cube Root Property

The cube root property is a useful tool when dealing with expressions involving both multiplication and division under a cube root. This property elaborates on how to simplify the derivative of these cube root expressions, laying the foundation for easier simplifications.
The property is expressed as:

• \( \frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}} \).

This tells us that we can simplify a fraction of cube roots by writing it as a single cube root of the fraction.
In our solved exercise example, this property was used to transition from two separate cube roots \( \frac{\sqrt[3]{48x^7}}{\sqrt[3]{6x}} \) into a single, simplified expression: \( \sqrt[3]{\frac{48x^7}{6x}} \).
By understanding and applying the cube root property, you can efficiently condense expressions and set the stage for further simplification steps, such as applying the quotient rule for exponents or recognizing perfect cubes. This property is fundamental for algebraic manipulation when dealing with cube roots.