Problem 64 - Intermediate Algebra

Question

Change each exponential expression to a radical. See Example 6. $$ \left(x^{3}+y^{3}\right)^{1 / 3} $$

Step-by-Step Solution

Answer

The expression \((x^3 + y^3)^{1/3}\) becomes \(\sqrt[3]{x^3 + y^3}\).

Step 1: Identify the Exponent as a Fraction

The given expression is \((x^3 + y^3)^{1/3}\). Notice that the exponent \(1/3\) is a fractional exponent.

Step 2: Recognize the Meaning of the Fractional Exponent

The expression \((x^3 + y^3)^{1/3}\) can be translated to a radical expression because the exponent \(1/3\) signifies a cube root. The numerator \(1\) is the power of the expression inside the radical, and the denominator \(3\) represents the root.

Step 3: Transform the Fractional Exponent into a Radical

Change the expression from \((x^3 + y^3)^{1/3}\) to the equivalent radical form. Since the denominator is \(3\), it becomes \(\sqrt[3]{x^3 + y^3}\), which is the cube root of the expression \(x^3 + y^3\).

Key Concepts

Radical Expressions

A radical expression involves roots, such as square roots, cube roots, or even higher roots. You often see them written with the radical symbol (√), where the number inside is called the radicand. Radical expressions can also include a small number, called the index, just outside and to the top left of the radical symbol, indicating the degree of the root. For example, if you see \(\sqrt[3]{a} \), it refers to the cube root of \( a \). The whole idea of a radical expression is about finding a number, which when multiplied by itself a certain number of times, gives the original radicand.

When working with radical expressions, it's crucial to understand how to switch between radical form and exponent form. This helps in simplifying algebraic expressions and solving equations that involve different mathematical operations. In many cases, radical expressions confirm the equality or relationship between radical and exponential forms.

Cube Root

The cube root is an example of a special type of radical expression. It involves finding a number which when multiplied by itself three times yields the original number or expression. In mathematical terms, the cube root of a number \( a \) is a number \( b \) such that \( b^3 = a \).

Cube roots can be denoted using the radical symbol with an index of 3, like this: \(\sqrt[3]{a}\). The process of finding a cube root is known as extracting the cube root, and it plays a vital role in algebra and geometry—particularly when dealing with volumes or solving cubic equations.

It's often easier to express cube roots using fractional exponents when performing calculations, because it allows for simpler algebraic manipulation. For example, the cube root of 8, denoted as \( 8^{1/3} \), equals 2 because \( 2^3 = 8 \).

Exponential Expressions

Exponential expressions are expressions that involve exponents, where a base is raised to a certain power. These powers, or exponents, are the number of times the base is multiplied by itself. A typical exponential form is \( a^n \), where \( a \) is the base and \( n \) is the exponent.

In cases with fractional exponents, the expression represents both a power and a root. The numerator of the fraction is the power, while the denominator signifies the type of root. For example, \( x^{1/3} \) is the same as taking the cube root of \( x \). This concept allows for flexibility and versatility when solving equations and simplifying complex expressions.

Understanding exponential expressions helps to bridge the gap between different forms of mathematical expressions. It serves as a foundation for higher-level math topics, such as calculus and complex numbers, where the properties of exponents are extensively used.