Problem 53 - Algebra 2

Question

Simplify. $$ \frac{1}{\sqrt[3]{10}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{\sqrt[3]{100}}{10} \).

Step 1: Identify the Expression

The given expression is \( \frac{1}{\sqrt[3]{10}} \). We need to simplify this expression, which involves rationalizing the denominator.

Step 2: Multiply by the Conjugate

To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt[3]{10^2} \). This gives:\[\frac{1}{\sqrt[3]{10}} \times \frac{\sqrt[3]{10^2}}{\sqrt[3]{10^2}} = \frac{\sqrt[3]{10^2}}{\sqrt[3]{10^3}}\].

Step 3: Simplify the Denominator

Simplify \(\sqrt[3]{10^3}\). Since the cube root and the cube cancel each other out, \(\sqrt[3]{10^3} = 10\).

Step 4: Simplify the Expression

Substitute back to obtain the final simplified form:\[\frac{\sqrt[3]{10^2}}{10}\], which simplifies to:\[\frac{\sqrt[3]{100}}{10}\].

Key Concepts

Simplifying expressions

Simplifying expressions involves reducing complex mathematical expressions to their simplest form. This process often makes it easier to work with equations and assess their true value. Simplification might include removing unnecessary elements or restructuring the expression for clarity. In the context of cube roots and rationalizing denominators, simplifying can involve eliminating radicals from the denominator, thereby making calculations more straightforward.

To simplify an expression, follow these tips:

• Identify like terms or equal parts that can be combined.
• Factor numbers and expressions to uncover hidden simplifications.
• Consider the properties of roots and powers to see if they simplify easily.

By simplifying, we make expressions more comprehensible and manageable for further calculations. This is especially useful in algebra, where you're often solving for variables. It also provides a clearer picture when checking for equivalence or incompatibilities within mathematical statements.

Cube roots

A cube root is a special type of root where you are looking for a number that, when multiplied by itself twice, results in the original number. For instance, the cube root of 8 is 2 because multiplying 2 by itself twice (2 x 2 x 2) equals 8.

Cube roots are denoted with a radical symbol (∛) and the operation is akin to finding the side of a cube given its volume – a valuable concept in geometry and algebra.

Understanding cube roots involves:

• Recognizing numbers that are perfect cubes like 1, 8, 27, etc.
• Applying cube root extractions to simplify complex expressions.
• Using cube root rules to help in solving equations.

When working with cube roots in expressions, especially in fractions, it often comes with the challenge of rationalizing denominators. This means altering the expression to avoid radicals in the denominator while keeping the expression equivalent.

Fraction operations

Working with fractions is a skill used across various branches of mathematics, from simple arithmetic to complex algebra. Operations involving fractions include addition, subtraction, multiplication, and division.

In the realm of algebra, handling fractions efficiently often hinges on understanding how to manipulate the numerator and the denominator. This is crucial when working with expressions that have cube roots in them.

Key points in fraction operations include:

• Finding a common denominator when adding or subtracting fractions.
• Multiplying fractions by multiplying across numerators and denominators.
• Dividing one fraction by another by multiplying by the reciprocal.
• Rationalizing denominators to remove roots or complex numbers.

To rationalize a denominator, particularly when it involves cube roots, multiply both the numerator and denominator by an expression that will eliminate the root from the denominator. This is essential for expressions to be in their simplest, most workable form.