Problem 26 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{y^{17}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression √(y^17). Answer: y^(17/2)

Step 1: Identify the terms inside the radical

We have the term \(y^{17}\) inside the square root. We will simplify this expression by applying the rules for radicals.

Step 2: Apply the rule for radicals

We will use the rule that states that \(\sqrt{a^m} = a^{\frac{m}{n}}\) for taking the nth root of a power. In this case, we are taking the square root, so n = 2. Apply this rule to the expression: $$ \sqrt{y^{17}} = y^{\frac{17}{2}} $$

Step 3: Final result

After simplifying the expression, we are left with: $$ y^{\frac{17}{2}} $$ This is our final simplified expression for the given radical expression.

Key Concepts

Simplifying Radicals

Simplifying radicals is about reducing them to their simplest possible form. This process involves using certain rules and techniques to make expressions easier to work with. The goal is to remove any perfect squares or cubes from under the radical sign whenever possible. When you have expressions like \( \sqrt{y^{17}} \), it is important to simplify them avoid any unnecessary complexity.

• Understand the type of root you are dealing with. In this case, it is a square root.
• Use the properties of exponents to simplify the expression under the radical.

By systematically applying these techniques, you can make radical expressions much more manageable and less intimidating.

Exponents and Powers

Exponents and powers are at the heart of simplifying radicals. When working with expressions like \( y^{17} \), you are dealing with powers of numbers or variables. Exponents indicate how many times a number is multiplied by itself. For example, \( y^{17} \) means \( y \) is multiplied by itself 17 times.

• Remember that raising a power to another power involves multiplication of the exponents.
• When simplifying expressions under a radical, you'll often need to rewrite the exponents in a different form, such as a fraction.

Understanding these concepts helps simplify the expression \( \sqrt{y^{17}} \) because it allows you to break down the exponent into a form that works with the radical.

Mathematical Rules

There are specific mathematical rules that aid in simplifying radicals exemplified by \( \sqrt{y^{17}} \). One such rule is the rule for nth roots. For any base \( a \) raised to the power \( m \), the nth root is expressed as \( \sqrt[n]{a^m} = a^{m/n} \).

• This rule helps convert a root into a fraction, which makes it simpler to handle.
• Applying this rule makes it easier to perform operations on powers.
• It’s the fundamental principle used to simplify radicals like \( \sqrt{y^{17}} \).

By applying this rule, you are effectively able to transform and then simplify expressions involving powers and radicals.

Square Root Simplification

Square root simplification is a common operation that makes working with expressions easier. The aim is to express the square root in its simplest form using the rules of exponents. In our example \( \sqrt{y^{17}} \), you need to recognize that the square root implies a power of \( \frac{1}{2} \).

• Convert the square root by changing the radical expression into a rational exponent, turning \( \sqrt{y^{17}} \) into \( y^{\frac{17}{2}} \).
• The resulting expression, \( y^{\frac{17}{2}} \), is considered simpler because it uses fewer symbols and is straightforward.
• It's important to verify that all steps followed properly simplify the expression without any mistakes.

Understanding how square root simplification works can greatly ease the process of manipulating and understanding more complex radical expressions.