Problem 8 - Algebra and Trigonometry

Question

Write each radical expression using exponents, and each exponential expression using radicals. $$ \sqrt[3]{7^{2}} $$

Step-by-Step Solution

Answer

\( \sqrt[3]{7^2} \) as an exponent is \( 7^{2/3} \), and vice versa.

Step 1: Identify the Radical Expression

We start by identifying the given expression: \( \sqrt[3]{7^2} \). This is a cube root of 7 squared.

Step 2: Write Radical as an Exponent

The expression \( \sqrt[3]{a} \) can be written as \( a^{1/3} \) in terms of exponents. Therefore, \( \sqrt[3]{7^2} \) can be written as \( (7^2)^{1/3} \).

Step 3: Apply Power of a Power Rule

According to the power of a power rule in exponents, \( (a^m)^n = a^{m\cdot n} \). Apply this rule to \( (7^2)^{1/3} \) to get \( 7^{2 \cdot 1/3} = 7^{2/3} \).

Step 4: Convert the Exponential Expression to Radical Form

Now, let's express \( 7^{2/3} \) back into radical form to ensure the conversion is correct. According to rules of radicals, \( a^{m/n} \) can be expressed as \( \sqrt[n]{a^m} \). Thus, \( 7^{2/3} = \sqrt[3]{7^2} \), verifying our steps were executed correctly.

Key Concepts

Cube Root

A cube root is a type of radical expression that involves finding the number which, when multiplied by itself three times, gives the original number. For example, in the expression \(\sqrt[3]{7^2}\), the cube root indicates the operation to find one of three equal factors of the product, \(7^2\).

This cube root expression tells us to find a number, which when used thrice as a factor, equals \(7^2\). The cube root is crucial in various mathematical contexts, especially in simplifying and solving equations. Understanding cube roots helps in grasping the concept of radicals in dealing with cubic and higher-degree equations in algebra.

Exponent Rules

Exponent rules are essential when simplifying expressions and performing operations involving powers. The power of a power rule is particularly useful here. This rule states that when raising a power to another power, you multiply the exponents. For example, \((a^m)^n = a^{m \cdot n}\).

Applying this rule to the expression \((7^2)^{1/3}\), we multiply the exponents to simplify the expression into \(7^{2/3}\). Exponent rules allow us to handle complex expressions and break them down into simpler parts to better understand and solve them.

Conversion Between Radicals and Exponents

The conversion between radicals and exponents helps transition between different forms of mathematical expressions with ease. Understanding this conversion is crucial because it allows you to solve equations more flexibly and interchangeably.

For instance, the expression \(\sqrt[3]{7^2}\) can be transformed to \((7^2)^{1/3}\) by converting the cube root to an exponent. This is effective in simplifying expressions further to \(7^{2/3}\).

• Radicals to exponents: \(\sqrt[n]{a^m} = a^{m/n}\)
• Exponents to radicals: \(a^{m/n} = \sqrt[n]{a^m}\)

Mastering this conversion equips you with the versatility to solve problems presented in either form efficiently.