Problem 91 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \sqrt{\sqrt{y}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( y^{\frac{1}{4}} \).

Step 1: Understand the Problem

We need to simplify the expression \( \sqrt{\sqrt{y}} \) and express it using rational exponents. Given that all variables are positive, we can work with the properties of exponents and radicals safely.

Step 2: Express Internal Square Root with Exponents

The expression \( \sqrt{y} \) can be rewritten using exponents. A square root of a number or expression is equivalent to raising that number or expression to the power of \( \frac{1}{2} \). Therefore, \( \sqrt{y} = y^{\frac{1}{2}} \).

Step 3: Apply the Radical to the Exponent Form

Given \( \sqrt{\sqrt{y}} \), we substitute \( \sqrt{y} \) with \( y^{\frac{1}{2}} \). Hence, the expression becomes \( \sqrt{y^{\frac{1}{2}}} \).

Step 4: Convert the Outer Square Root to an Exponent

Apply the same principle of converting the square root to an exponent. The square root is equivalent to raising the expression inside to the power of \( \frac{1}{2} \). Now, \( \sqrt{y^{\frac{1}{2}}} = (y^{\frac{1}{2}})^{\frac{1}{2}} \).

Step 5: Simplify Using the Power of a Power Rule

Use the power of a power rule, which states that \( (x^m)^n = x^{m \cdot n} \). Hence, \( (y^{\frac{1}{2}})^{\frac{1}{2}} = y^{\frac{1}{2} \cdot \frac{1}{2}} = y^{\frac{1}{4}} \).

Step 6: Final Step: Write the Simplified Expression

The simplified expression in terms of rational exponents is \( y^{\frac{1}{4}} \).

Key Concepts

Exponential Notation

Exponential notation is a mathematical way of representing numbers or expressions that involve repeated multiplication. It's a concise and straightforward method to express large numbers or repeated operations. This notation makes computation simpler and more manageable, allowing us to apply various mathematical rules effectively.

• The base, such as "y" in an expression, tells us what number is being multiplied.
• The exponent, such as "n" in the expression "y^n", indicates how many times the base is multiplied by itself.

An expression using exponents, like \( y^{\frac{1}{2}} \), tells us that we take the square root of \( y \). This is because raising \( y \) to the power of \( \frac{1}{2} \) is equivalent to finding its square root. Exponential notation provides a helpful framework to work with roots and powers, especially when dealing with complex mathematical operations.

Simplifying Expressions

Simplifying expressions is essential for making them easier to work with in equations and calculations. This process involves finding an equivalent expression that is more straightforward or is presented in a standardized form. In mathematic terms, simplifying can involve:

• Combining like terms.
• Using foundational algebraic rules to simplify the operations involved.

For example, to simplify \( \sqrt{\sqrt{y}} \), we first recognize that the double square root is just a repeated use of the square root function. Using exponents, \( \sqrt{y} \) can be expressed as \( y^{\frac{1}{2}} \). Therefore, the original expression \( \sqrt{\sqrt{y}} \) becomes \( \sqrt{y^{\frac{1}{2}}} = (y^{\frac{1}{2}})^{\frac{1}{2}} \). By rewriting the expression in this way, we can apply rules for exponents to simplify it further.

Power of a Power Rule

The power of a power rule is a fundamental exponent rule in math that makes simplifying nested exponents straightforward. This rule states that when you take an exponent to another exponent in the form \( (x^m)^n \), it can be simplified by multiplying the exponents to give \( x^{m \cdot n} \).

• For example, if you have \( (y^{\frac{1}{2}})^{\frac{1}{2}} \), use the power of a power rule to get \( y^{\frac{1}{2} \times \frac{1}{2}} \).
• This simplifies to \( y^{\frac{1}{4}} \).

By applying this rule, we can simplify expressions like \( \sqrt{\sqrt{y}} \) efficiently, providing a clearer and more concise expression of the problem. This rule is particularly useful when dealing with repeated roots or powers, as it allows for prompt and precise simplification.