Problem 59 - Elementary and Intermediate Algebra

Question

Change each radical to an exponential expression. See Example 5. $$ \sqrt[3]{a^{2}-b^{2}} $$

Step-by-Step Solution

Answer

\( (a^2 - b^2)^{1/3} \)

Step 1: Identify the Radical and Its Index

The given expression is \( \sqrt[3]{a^2 - b^2} \). This is a cube root, which means the radical has an index of 3. Exponentially, this means we will raise the expression \( a^2 - b^2 \) to the power of \( \frac{1}{3} \).

Step 2: Express the Radical as an Exponent

Convert the cube root into an exponent. The expression inside the radical is \( a^2 - b^2 \), and the radical index is 3. Therefore, the exponential form of \( \sqrt[3]{a^2 - b^2} \) is \( (a^2 - b^2)^{1/3} \).

Key Concepts

Understanding Radicals

Radicals are expressions that involve roots, such as square roots or cube roots. In mathematics, a radical sign \( \sqrt[n]{} \) denotes the root of a number or expression. The number \( n \) inside the radical sign is known as the index. For example, in a square root \((\sqrt{})\), the index is 2, which is typically not written. For a cube root \((\sqrt[3]{})\), the index is 3.

Radicals are used to find the number which, when multiplied by itself \( n \) times, results in the original number. For instance:

• \( \sqrt[3]{8} = 2 \) because \( 2^3 = 8 \).
• \( \sqrt[4]{16} = 2 \) because \( 2^4 = 16 \).

They are useful in simplifying complex expressions or solving equations where direct computation would be complex. Converting radicals to exponential form can often make it easier to manipulate or combine with other algebraic expressions.

Decoding the Cube Root

A cube root is a specific type of radical that finds the number which, when multiplied by itself three times, equals the original number. The cube root of a number \( x \) is represented mathematically as \( \sqrt[3]{x} \). It seeks a value \( y \) such that \( y^3 = x \).

For example:

• \( \sqrt[3]{27} = 3 \)
• \( \sqrt[3]{64} = 4 \)
• \( \sqrt[3]{125} = 5 \)

Cubes and cube roots are especially important when dealing with polynomial equations or geometry involving three dimensions. In practical applications, taking the cube root helps solve problems related to volume and density. When converting cube roots to exponential expressions, we use an exponent of \( \frac{1}{3} \). This notation simplifies the cube root into a more algebraically friendly format, making it easier to perform operations like multiplication and division.

Basics of Exponents

Exponents provide a method of representing repeated multiplication of a number by itself. The expression \( a^n \) denotes \( a \) being multiplied by itself \( n \) times. Exponents can also be fractional, representing roots as an inverse operation to powers.

When dealing with radicals, they can be rewritten with fractional exponents. Specifically:

• The square root \( \sqrt{a} \) can be expressed as \( a^{1/2} \).
• The cube root \( \sqrt[3]{a} \) becomes \( a^{1/3} \).

This conversion allows for easier manipulation within algebraic expressions. It is particularly useful when combining terms or applying rules of exponents, such as:\( a^m \cdot a^n = a^{m+n} \) or \((a^m)^n = a^{mn}\).

Understanding exponents and the capability to shift between radical and exponential forms significantly enhances problem-solving aptitude in algebra.