Problem 103 - Algebra 2

Question

Write each expression in radical form. $$ \left(16 w^{3}\right)^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

The expression \( \left(16 w^{3}\right)^{\frac{1}{2}} \) in radical form is \( 4w^{3/2} \)

Step 1: Identify the terms inside the parentheses

Here, we can see that the expression inside the parentheses is 16w⁶. The base numbers to be rooted are 16 and w⁶.

Step 2: Apply the square root

The exponent of 1/2 is equivalent to the square root. So, we take the square root of both 16 and w^(3). The square root of 16 is 4, and the square root of w^(3) cannot be simplified, so it is written as w^(3/2).

Step 3: Write the Final Answer

Join the rooted 16 and w^(3) together to form the final expression, giving 4w^(3/2).

Key Concepts

Exponents and Radicals

The connection between exponents and radicals is a crucial concept in algebra. An exponent represents how many times a number, known as the base, is multiplied by itself. Radicals, on the other hand, involve finding the root of a number. For example, the square root and cube root are common types of radicals. Mathematically, radicals can often be expressed using fractional exponents. This means that the expression \((16 w^{3})^{\frac{1}{2}}\) in the exercise involves applying the rules of exponents and radicals together.
The exponent \(\frac{1}{2}\) is a special type, which indicates a square root. Thus, when you see \(a^{\frac{1}{2}}\), it's the same as saying \(\sqrt{a}\). In algebra, understanding this relationship will help you convert expressions between radical and exponential forms quickly.

• Radicals are just another way to express fractional exponents.
• The exponent \(\frac{1}{2}\) signifies taking the square root.
• Converting between radicals and exponents is essential in simplifying expressions.

By recognizing this relationship, you'll more easily tackle algebraic problems.

Simplifying Expressions

To simplify expressions, we break them down into their simplest form. Simplifying often involves reducing or eliminating exponents, combining like terms, or factoring. This process makes the expression easier to work with. Take the exercise, \(\left(16 w^{3}\right)^{\frac{1}{2}}\), for instance. To simplify it, we follow certain steps that involve both exponents and radicals.
The expression involves a square root, indicated by the exponent \(\frac{1}{2}\). We first identify the parts of the expression: 16 and \(w^6\). The next step involves simplifying by taking the square root of each component.
These steps are:

• Find the square root of 16, which is 4.
• Calculate the square root of \(w^6\), resulting in \(w^{3/2}\).

The result is a simplified expression: \(4w^{3/2}\). This represents how we can transform an expression into a more straightforward form by understanding the properties of exponents and radicals.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators. They form the foundation of algebra and encompass concepts such as addition, subtraction, multiplication, division, and exponentiation. An algebraic expression like \((16 w^{3})^{\frac{1}{2}}\) combines these elements effectively.
Variables such as \(w\) stand in for unknown or variable quantities and are subjected to mathematical operations just like numbers. In the given exercise, \(w^{3}\) is a part of the expression raised to the power of \(\frac{1}{2}\), which involves both multiplication and radical concepts.

• Algebraic expressions often require manipulation using rules and properties of exponents and radicals.
• Variables and constants are integral parts of these expressions, allowing them to model real-world situations or complex mathematical ideas.
• Simplifying these expressions often leads to easier notation and understanding.

Understanding and simplifying algebraic expressions are key skills in algebra, enabling more complex problem-solving and analysis.