Problem 22 - Algebra 2

Question

Write each radical using rational exponents. $$ \sqrt[3]{62} $$

Step-by-Step Solution

Answer

The expression \( \sqrt[3]{62} \) is written as \( 62^{\frac{1}{3}} \) using rational exponents.

Step 1: Apply the radical to exponent conversion

The general rule is \(\sqrt[n]{a} = a^{1/n}\).

Step 2: Convert

\(\sqrt[3]{62} = 62^{1/3}\)

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics representing repeated multiplication of a number by itself. For instance, when we say \( a^3 \), it means the number \( a \) is multiplied by itself three times. Exponents are not limited to whole numbers; they can also be fractions. This concept allows us to perform operations on very large or small numbers efficiently.
By using exponents, calculations become simpler:

• \( a^1 \) is just \( a \) itself.
• \( a^0 = 1 \), provided \( a \) is not zero.
• Products of the same base are added, like \( a^m \times a^n = a^{m+n} \).

It’s essential to understand these basic rules, as they form the foundation for more complex operations like radicals and fractional exponents.

Radicals

Radicals involve roots, such as square roots or cube roots, and are the opposite of using exponents. The radical symbol \( \sqrt{} \) represents a root, with an index indicating which root.
For example:

• The square root of \( x \) is \( \sqrt{x} \), where no index is usually written, implying a square root.
• Cube roots, like \( \sqrt[3]{x} \), indicate the number that, when cubed, yields \( x \).

The transition to dealing with radicals more flexibly often involves converting them to exponents. Each radical expression can correspond to an expression with exponents.
Though initially appearing complex, recognizing how to switch between the two formats simplifies calculations significantly, making mathematics more manageable.

Fractional Exponents

Fractional exponents are a combination of exponents and radicals. They offer an alternative way to express root operations. For any radical expression \( \sqrt[n]{a} \), it can be written in terms of fractional exponents as \( a^{\frac{1}{n}} \).
This conversion is based on the definition of roots as fractional powers:

• \( \sqrt{a} = a^{\frac{1}{2}} \)
• \( \sqrt[3]{a} = a^{\frac{1}{3}} \)
• In a more general form, \( \sqrt[n]{a} = a^{\frac{1}{n}} \)

Fractional exponents allow easier manipulation and combination of radicals in equations since they adhere to all the rules of exponents. Understanding how to convert radicals to fractional exponents is crucial for simplifying expressions and solving equations efficiently in algebra.