Problem 10 - Precalculus: Mathematics for Calculus

Question

Write each radical expression using exponents, and each exponential expression using radicals. Radical expression = \(\sqrt[3]{7^{2}}\) Exponential expression = ?

Step-by-Step Solution

Answer

The exponential expression is \(7^{2/3}\).

Step 1: Understand the Radical Expression

The given radical expression is \( \sqrt[3]{7^2} \). Here, the number inside the radical sign is \(7^2\), and the radical is a cube root.

Step 2: Apply the Radical to Exponential Conversion Rule

To convert a radical expression to an exponential form, use the rule \( \sqrt[n]{a^m} = a^{m/n} \). For this problem, \(a = 7\), \(m = 2\), and \(n = 3\).

Step 3: Substitute Values into the Rule

Using the formula from the previous step, substitute the values: \( \sqrt[3]{7^2} = 7^{2/3} \).

Step 4: Finalize the Exponential Expression

The exponential expression that corresponds to the radical expression \( \sqrt[3]{7^2} \) is \(7^{2/3}\).

Key Concepts

Understanding Radical Expressions

Radical expressions are mathematical expressions that involve a root symbol, such as a square root or cube root. The root symbol looks like this: \(\sqrt{}\).
In the context of radical expressions, the number inside the root symbol is called the "radicand" and the number indicating the degree of the root, such as 2 for square roots and 3 for cube roots, is called the "index".
For example, in \(\sqrt[3]{7^2}\), 7 is the base number and 2 is the exponent, forming the radicand \(7^2\), and the index is 3, indicating a cube root.
Radicals are useful when you want to express repeated multiplication or division operations.

• The most common radicals are square roots (\(\sqrt{}\)) and cube roots (\(\sqrt[3]{}\)).
• These operations help simplify expressions and solve equations involving powers.

Explaining Cube Roots

A cube root is the number that, when multiplied by itself three times, gives the original number.
For example, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\). Cube roots are particularly useful when dealing with volumes, as they help reverse the cubing operation.
When a cube root is part of a radical expression, it is typically written using the symbol \(\sqrt[3]{}\).
This symbol indicates the third root or cube root of the number inside. Understanding cube roots helps in various mathematical processes, including simplification and solving cubic equations.

• Cube roots help you find the original side length of a cube when you know the volume.
• They also help simplify complex expressions by breaking them down into easily manageable pieces.

The Importance of Conversion Rules

Conversion rules play a vital role in translating between different mathematical expressions.
Specifically, the conversion rule \(\sqrt[n]{a^m} = a^{m/n}\) helps us convert radical expressions into exponential form.
This rule is crucial because exponential expressions can be easier to manipulate, especially in equations and calculus.
It allows one to express roots as power operations, which are more straightforward to handle.When working with the conversion of \(\sqrt[3]{7^2}\), we use the conversion rule to rewrite it in exponential form as \(7^{2/3}\).

• Exponential forms are generally easier to differentiate and integrate in calculus.
• This powerful rule transforms roots into fractions, offering a cleaner method to solve problems.