Problem 14 - Algebra 1: Concepts and Skills

Question

Rewrite the expression using radical notation. $$ 6^{1 / 3} $$

Step-by-Step Solution

Answer

So, the expression \(6^{1 / 3}\) can be written in radical notation as \(\sqrt[3]{6}\)

Step 1: Understanding Exponent Notation

Exponents are a way of showing repeated multiplication. In this expression \(6^{1/3}\), the number 6 is the base and the number \(1/3\) is the exponent. When an exponent is a fraction, the numerator (top number) represents the power, and the denominator (bottom number) represents the root.

Step 2: Converting to Radical Notation

In radical notation, the denominator of the exponent becomes the index of the radical, and the numerator of the exponent becomes the power of the radicand (the number under the radical). In this case, the denominator of the exponent is 3, so the index of the radical is also 3. The numerator of the exponent is 1, so the power of the radicand is 1, which we don't usually write. Therefore, the radical notation of \(6^{1/3}\) is \(\sqrt[3]{6}\).

Key Concepts

Understanding Exponents

Exponents are used to represent repeated multiplication of a number by itself. For example, in the simplified expression \(6^3\), 6 is the base, and 3 is the exponent.

• The base is the number we multiply.
• The exponent tells us how many times to multiply the base by itself.

Exponents can be whole numbers, fractions, or even negative numbers, each affecting the base differently. When dealing with fractional exponents, as in \(6^{1/3}\), we need to understand that the numerator denotes the power, and the denominator denotes the root. This means that fractional exponents are a compact way to express roots, such as the cube root, in this context. Whenever you see a fraction in the exponent, like \(1/3\) in \(6^{1/3}\), think about the numerator as a power and the denominator as a type of root to apply.

The Role of Roots

Roots are the inverse operation of exponents. They help us find the original base that was raised to an exponent to achieve a certain number. Common roots include square roots \(\sqrt{}\) and cube roots \(\sqrt[3]{}\). When you see a fractional exponent, it can be translated into a root notation. In the expression \(6^{1/3}\):

• The number 3 as the denominator tells us to take the cube root of 6.
• If the fraction was \(6^{m/n}\), it would mean the nth root of 6 raised to the power m.

Radical notation uses the radical sign (√) to signify roots. It helps transform the expression from its exponent form to a more visual format that highlights the index or type of root applied to the base number.

Understanding the Radicand

The radicand is the number or expression inside the radical sign that we perform the root operation on. In the notation \(\sqrt[3]{6}\), 6 is the radicand.

• The radicand can be any real number or algebraic expression.
• In radical notation, it's important because what we calculate the root of determines the final result.

The cube root in \(\sqrt[3]{6}\) tells us to find the number that gives 6 when it is cubed. This conversion from fractional exponent notation \(6^{1/3}\) to radical notation \(\sqrt[3]{6}\) allows us to view and solve problems involving exponents and roots in a more intuitive way. The radicand plays a central role as it is the basis of the root that we are solving for or working with.