Problem 29 - A Graphical Approach to College Algebra

Question

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$\sqrt{y \cdot \sqrt{y}}$$

Step-by-Step Solution

Answer

\(y^{3/4}\)

Step 1: Understand the expression

We start with the expression \(\sqrt{y \cdot \sqrt{y}}\). Note that it involves a square root and another square root inside it, both containing the variable \(y\). Our task is to express this using positive rational exponents.

Step 2: Rewrite inner square root as exponent

Inside the main square root, we have \(\sqrt{y}\). Recall that the square root of \(y\) can be written using rational exponents as \(y^{1/2}\). Therefore, the expression inside the square root becomes \(y \cdot y^{1/2}\).

Step 3: Simplify the expression inside the square root

To simplify \(y \cdot y^{1/2}\), we add the exponents of \(y\). The rules of exponents tell us that \(a^m \cdot a^n = a^{m+n}\). So, we have \(y^{1} \cdot y^{1/2} = y^{1 + 1/2} = y^{3/2}\).

Step 4: Rewrite the outer square root using exponents

Now that the expression inside the square root is \(y^{3/2}\), we take the square root of this by raising it to the power of \(1/2\). That is, \((y^{3/2})^{1/2}\).

Step 5: Apply the power of a power rule

Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we find that \((y^{3/2})^{1/2} = y^{3/2 \cdot 1/2} = y^{3/4}\).

Step 6: Write the final expression with positive rational exponents

The expression \(\sqrt{y \cdot \sqrt{y}}\) can be rewritten as \(y^{3/4}\) using positive rational exponents.

Key Concepts

Exponent Rules

Understanding exponent rules is essential when working with expressions involving powers and roots. These rules help us manipulate expressions by providing methods to simplify, multiply, and raise powers to other powers.

Here are some fundamental exponent rules to remember:

• Product of Powers Rule: When multiplying like bases, you add their exponents. For instance, \( a^m \cdot a^n = a^{m+n}\).
• Power of a Power Rule: When raising a power to another power, you multiply the exponents. For example, \( (a^m)^n = a^{m \cdot n} \).
• Power of a Product Rule: When raising a product to an exponent, raise each factor to the exponent. That is, \( (ab)^n = a^n \cdot b^n \).
• Fractional Exponents: A fractional exponent like \( a^{1/n} \) is equivalent to the \( n \)-th root of \( a \). Similarly, \( a^{m/n} = (a^m)^{1/n} \).

Having a solid grasp of these rules allows you to handle complex expressions more easily, and as seen in the original exercise, using these rules can greatly simplify the process of rewriting expressions with rational exponents.

Simplifying Expressions

Simplifying expressions involves making them as clean and straightforward as possible without changing their value. It's a crucial skill in algebra that helps in understanding the essence of the problem and making calculations more manageable.

Here's how you can approach simplifying expressions:

• Identify Components: Look for components like exponentials, roots, products, and constants within the expression.
• Apply Exponent Rules: Use exponent laws such as the product and power rules to combine like terms and simplify exponentials.
• Combine Like Terms: Add or subtract terms that have the same variables and exponents to reduce the expression.
• Rewrite Roots as Exponents: Transform roots into exponents to easier manipulate and simplify them, as done in the original exercise where roots were converted into fractional exponents.

By following these steps, you can simplify complex algebraic expressions efficiently—such as turning \( \sqrt{y \cdot \sqrt{y}} \) into \( y^{3/4} \). Each step breaks down the simplification process, making it easier to comprehend and execute.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a particular mathematical quantity. They can be as simple as \( x \) or involve complex combinations like \( 3x^2 + 2xy - 5x + 7 \).

Key components of algebraic expressions to understand include:

• Variables and Constants: Variables (like \( x \) and \( y \)) are symbols that stand in for numbers. Constants are fixed values like \( 5 \) or \( 3 \).
• Terms: A term is a part of the expression separated by addition or subtraction. For instance, in \( 3x^2 + 2xy - 5 \), each piece is a term.
• Coefficients: A coefficient is a number multiplying a variable (e.g., \( 3 \) in \( 3x \)).
• Expressions vs. Equations: Remember, an expression does not have an "equals" sign, while an equation does.

Understanding these basics helps you manipulate algebraic expressions and solve problems like the one provided by transitioning from nested roots to simpler forms using rational exponents.