Problem 64 - Brief Applied Calculus

Question

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \sqrt[3]{\frac{8}{x^{6}}} $$

Step-by-Step Solution

Answer

The expression is \( 2x^{-2} \) in power form.

Step 1: Understand the Cube Root

The cube root expression is given as \( \sqrt[3]{\frac{8}{x^{6}}} \). It can be rewritten in power form by using the property of radicals: \( \sqrt[n]{a} = a^{1/n} \). So, \( \sqrt[3]{\frac{8}{x^{6}}} = (\frac{8}{x^{6}})^{1/3} \).

Step 2: Distribute the Exponent

Using the property of exponents that \( (\frac{a}{b})^n = \frac{a^n}{b^n} \), apply the \( \frac{1}{3} \) power to both the numerator and the denominator, resulting in: \( (\frac{8}{x^6})^{1/3} = \frac{8^{1/3}}{(x^6)^{1/3}} \).

Step 3: Simplify the Numerator

Calculate \( 8^{1/3} \). Since \( 8 = 2^3 \), \( 8^{1/3} = (2^3)^{1/3} = 2^{3 \times \frac{1}{3}} = 2^1 = 2 \).

Step 4: Simplify the Denominator

Calculate \( (x^6)^{1/3} \). Use the property of exponents \((a^m)^n = a^{m \times n} \). So, \( (x^6)^{1/3} = x^{6 \times \frac{1}{3}} = x^2 \).

Step 5: Write the Final Expression in Power Form

Combine the simplified numerator and denominator to rewrite the expression as a power of \( x \). The expression becomes \( \frac{2}{x^2} \). Since \( x^{-n} = \frac{1}{x^n} \), the final power form is \( 2x^{-2} \).

Key Concepts

Cube Roots

A cube root is an operation that finds a number which, when multiplied by itself three times, yields the original number. For example, the cube root of 8 is 2, since \(2 \times 2 \times 2 = 8\). The notation for a cube root is \(\sqrt[3]{a}\), where 3 is called the index. To rewrite a cube root into a more convenient form for mathematical operations, we use exponents. This conversion is based on the property that any root of a number can be expressed as a power, where:

• \(\sqrt[n]{a} = a^{1/n}\)

This shift from root notation to power notation simplifies complex arithmetic, especially when dealing with variables and higher-level expressions. By rewriting \(\sqrt[3]{\frac{8}{x^{6}}}\) as \((\frac{8}{x^{6}})^{1/3}\), you can manage and manipulate the components individually, allowing for easier integration into other mathematical processes.
Cube roots, like square roots, are fundamental in finding solutions to polynomial equations and are heavily used in geometry and other fields of mathematics.

Properties of Exponents

Exponents are a shorthand notation for repeated multiplication, and understanding their properties is crucial in manipulating algebraic expressions. Several properties help simplify expressions that involve exponents and roots.

• Product of powers: \(a^m \times a^n = a^{m+n}\)
• Power of a power: \((a^m)^n = a^{m \times n}\)
• Quotient of powers: \(\frac{a^m}{a^n} = a^{m-n}\) if \(a eq 0\)
• Power of a quotient: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)

These properties are used to simplify expressions effectively. For example, in the expression \((\frac{8}{x^{6}})^{1/3}\), we apply the power of a quotient property, rewriting it as \(\frac{8^{1/3}}{(x^6)^{1/3}}\). By segmenting the expression into a part for the numerator and a part for the denominator, each can be calculated separately using basic exponent rules with greater ease.

Conversion from Radical to Exponent Form

Transforming expressions from radical to exponent form can considerably simplify their evaluation and manipulation. Radicals and exponents are two ways to express the same mathematical concept, specifically dealing with the roots of a number. This duality allows us to choose the form that is most convenient for the problem at hand.
For instance, in mathematics:

• The nth root of a number can be expressed as \(a^{1/n}\).

Thus, converting \(\sqrt[3]{\frac{8}{x^{6}}}\) into exponent form gives \((\frac{8}{x^{6}})^{1/3}\). This method simplifies complex roots into a form that intertwines naturally with algebraic operations like multiplication and division. By harnessing basic exponent rules, expressions become much more manageable. Consequently, exponent form is preferred in higher mathematics and calculus where complex operations require elegance and simplicity.

Simplifying Expressions

The process of simplifying expressions involves reducing them to their most basic form to make them easier to work with. This often means using properties of exponents or algebraic manipulations to combine like terms and eliminate any unnecessary components.
In our exercise, the expression \((\frac{8}{x^{6}})^{1/3}\) can be simplified by addressing both the numerator and the denominator individually. First, calculate cube root of 8, which is \(2\) (since \(2^3 = 8\)), resulting in \(2^{1}\). Then for the denominator, use the power of a power rule to find \((x^6)^{1/3} = x^2\).
Putting it all together, we simplify the expression to \(\frac{2}{x^2}\). Finally, using properties of exponents, rearrange this into a simpler expression, \(2x^{-2}\), which is easier for further manipulation in algebraic problems. It's about recognizing efficiency in calculations, a necessary skill in solving complex mathematical equations effectively and accurately.