Problem 73 - Intermediate Algebra

Question

Simplify each radical. Assume that all variables represent positive real numbers. $$ -\sqrt[3]{\frac{z^{21}}{27 x^{3}}} $$

Step-by-Step Solution

Answer

The simplified form is \(-\frac{z^7}{3x}\).

Step 1: Introduce Problem

We need to simplify the expression \(-\sqrt[3]{\frac{z^{21}}{27 x^{3}}}\) by breaking it down into simpler components.

Step 1: Simplify the Cube Root of the Denominator

The denominator inside the cube root is \(27x^3\). Since \(27 = 3^3\) and \(x^3\) is already a cube, \(\sqrt[3]{27x^3}\) simplifies to \(3x\).

Step 2: Simplify the Cube Root of the Numerator

The numerator inside the cube root is \(z^{21}\). Since \(z^{21} = (z^7)^3\), the cube root of \(z^{21}\) simplifies to \(z^7\).

Step 3: Combine the Simplified Expressions

Now, put together the simplified forms of both the numerator and the denominator. We have \(-\frac{z^7}{3x}\) since the negative sign outside the cube root applies to the entire expression.

Key Concepts

Cube Roots

A cube root is an operation that focuses on finding a number which, when multiplied by itself three times, equals the original number. The cube root is denoted by a small three inside the radical sign, like this: \(\sqrt[3]{...}\). It is different from a square root because it involves three factors instead of two.To fully grasp cube roots, remember the main goal is to "undo" a number raised to the third power. For instance, in the context of our exercise, we need to find the cube root of \(27x^3\). Since both components, \(27\) and \(x^3\), are perfect cubes (meaning they can be rewritten as a base number raised to the power of three), we can simplify them easily. - \(27 = 3^3\), so \(\sqrt[3]{27} = 3\).- \(x^3\) is already a cubic expression, so \(\sqrt[3]{x^3} = x\).Thus, the cube root of \(27x^3\) becomes \(3x\). The concept centers around reversing the exponentiation of three, making calculations straightforward once you identify which parts of the expression are cubes.

Radical Expressions

A radical expression includes a radical sign (\(\sqrt{\;}\)) and something inside, which is called the radicand. Radical expressions can involve square roots, cube roots, or higher roots.When simplifying radical expressions, the goal is to express them in their simplest form. This involves removing any perfect cubes (in the case of cube roots) or squares (for square roots) from under the radical sign as much as possible.In our exercise, the expression we are simplifying is \(-\sqrt[3]{\frac{z^{21}}{27 x^{3}}}\). Simplifying the numerator \(z^{21}\) involves rewriting it as a cube, which is \((z^7)^3\). Thus, \(\sqrt[3]{z^{21}} = z^7\), and similarly simplifying the denominator, \(\sqrt[3]{27x^3}\) results in \(3x\).Finally, combining these results gives us the simplified radical expression \(-\frac{z^7}{3x}\). This type of understanding helps make complex expressions more digestible.

Positive Real Numbers

The assumption that all variables in the expression are positive real numbers simplifies calculations significantly. In mathematics, real numbers include both rational numbers (like 5 or 0.75) and irrational numbers (like \(\pi\) or \(\sqrt{2}\)). Positive real numbers are simply the subset of these real numbers that are greater than zero.This assumption is crucial because it ensures that there are no surprises such as negative roots or undefined expressions. For instance, when we take the cube root of \(z^{21}\), assuming \(z\) is positive, guarantees that the result \(z^7\) remains straightforward and doesn't involve complex numbers.In our given problem, asserting positivity allows us to simplify across radical expressions without additional conditions or adjustments. This permits us to treat cubes and roots in a linear fashion, maintaining clarity throughout the simplification process.