Question

Write \(\sqrt[3]{9}\) as an expression with an exponent. A. \((\sqrt[3]{9})^1\) B. \(9^{\frac{1}{3}}\) C. \(3^1\) D. \(9^3\)

Short answer

B. \(9^{\frac{1}{3}}\)

Step-by-step solution

Understanding the problem

We need to write \(\sqrt[3]{9}\) as an expression with an exponent.

Identify the Radical Form

The given radical form is \(\sqrt[3]{9}\). Here, the '3' is the index, and '9' is the radicand.

Convert Radical to Exponent

To convert a radical to an exponent, use the rule \(\sqrt[n]{a} = a^{\frac{1}{n}} \). In this case, \(\sqrt[3]{9} = 9^{\frac{1}{3}}\).

Match with the Given Choices

Now compare the converted form \(9^{\frac{1}{3}}\) with the given options. The correct option is B. \(9^{\frac{1}{3}}\).

Key concepts

radical expressions

Radical expressions often seem intimidating, but they can be easier to work with once you understand the basics. A radical expression involves roots, such as square roots or cube roots. For example, the cube root of 9 is written as \( \sqrt[3]{9} \).

The number inside the root symbol is called the radicand (in this case, 9). The small number outside, which indicates the degree of the root, is the index (here, 3). Understanding this basic structure is essential in learning how to manipulate and simplify radical expressions.

exponent rules

Exponent rules are the foundation of simplifying many algebraic expressions, including converting radicals to exponent form. When you see a radical, it can be rewritten using exponents to make calculations easier.

For example, the general rule for converting a radical expression to an exponent form is \[ \sqrt[n]{a} = a^{\frac{1}{n}} \]. This allows you to treat the radical as a fractional exponent. In the exercise above, the cube root of 9 can be written as \ 9^{\frac{1}{3}} \ \.

Remembering a few key exponent rules will make your work much smoother:

• When multiplying like bases, add their exponents: \ a^m \cdot a^n = a^{m+n} \
  
• When dividing like bases, subtract their exponents: \ \frac{a^m}{a^n} = a^{m-n} \
  
• Any number raised to the power of zero is 1: \ a^0 = 1 \
  
• When raising a power to a power, multiply the exponents: \ (a^m)^n = a^{mn} \

mathematical notation

Having a solid understanding of mathematical notation can greatly assist in interpreting and solving algebra problems. Mathematical notation is a system of symbols and signs used to represent numbers, operations, relations, and grouping.

In the case of exponents and radicals, knowing how to correctly interpret and use these symbols is crucial. For example, the radical symbol \( \sqrt{} \) signifies the root of a number, while the exponent \( a^n \) indicates that a number is raised to a specific power.

Being familiar with the notation helps you read problems accurately and understand the steps required to solve them. For instance, recognizing that \ 9^{\frac{1}{3}} \ means the same as \( \sqrt[3]{9} \) can simplify your work and make you more efficient in solving algebraic problems. Learn the symbols well, and practice using them in different contexts to become proficient.