Problem 3 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ 27^{1 / 3} $$

Step-by-Step Solution

Answer

The expression in radical form is \(\sqrt[3]{27}\), which simplifies to 3.

Step 1: Understand the Exponent

The expression given is \(27^{1/3}\). The exponent \(1/3\) indicates that we need to take the cube root of 27. When you see a fractional exponent like this, the numerator represents a power, and the denominator represents a root. Here, since the numerator is 1, we are only concerned with the cube root.

Step 2: Rewrite in Radical Notation

Using the rule for fractional exponents, \(a^{m/n} = \sqrt[n]{a^m}\), we rewrite \(27^{1/3}\) as \(\sqrt[3]{27}\). This means we're looking for a number that, when multiplied by itself three times, results in 27.

Step 3: Simplify the Radical

Think about what number cubed equals 27. We know that \(3 \times 3 \times 3 = 27\). Therefore, the cube root of 27 is 3. This means \(\sqrt[3]{27} = 3\).

Key Concepts

Fractional Exponents

Fractional exponents, like the one seen in the expression \(27^{1/3}\), are not as complicated as they might seem. They provide a way to express roots using powers. In a fractional exponent \(a^{m/n}\), the numerator \(m\) acts as a power, and the denominator \(n\) indicates the root. This means that \(a^{1/3}\) translates to the cube root of \(a\). Using fractional exponents is beneficial because it allows us to switch easily between powers and roots, offering a concise way to represent calculations.

• The base number here is 27.
• The exponent is \(1/3\), meaning a cube root since 1 is the power and 3 is the root.

It is crucial to remember that fractional exponents not only provide an alternative to radical notation but also simplify algebraic operations by abiding to exponent rules.

Cube Root

The cube root is an important mathematical operation that focuses on finding a number which, when multiplied by itself three times, gives the original number. In radical notation, this is represented as \(\sqrt[3]{a}\), where \(a\) is the number you're taking the cube root of.
To solve \(\sqrt[3]{27}\), you need to determine what number cubed equals 27. We know that 3 cubed (\(3 \times 3 \times 3\)) equals 27. Therefore, the cube root of 27 is simply 3.

• \(\sqrt[3]{27} = 3\) because \(3^3 = 27\).
• Cube roots are the inverse operation of cubing a number.

Understanding cube roots helps in solving equations and simplifying expressions involving radical notation and fractional exponents.

Simplifying Radicals

Simplifying radicals involves reducing the radical expression to its simplest form. In the case of \(\sqrt[3]{27}\), it's already given in a form that can be simplified easily. Simplifying requires finding the cube root, which as we found, is 3.
Here are quick steps you might follow for simplifying cube roots:

• Determine if the number is a perfect cube (like 27).
• If the number is a perfect cube, find the integer that, when cubed, results in the number under the radical.

In our case, 27 is a perfect cube as \(3^3 = 27\). Thus, \(\sqrt[3]{27} = 3\).
Simplifying radicals is crucial for reducing algebraic expressions into their simplest forms, helping in understanding and solving problems involving radicals more effectively.