Problem 54 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{y-3} \sqrt{(y-3)^{5}} $$

Step-by-Step Solution

Answer

Answer: The product of the given expressions is \((y-3)^{\frac{11}{2}}\).

Step 1: Identify the terms and exponents

We have two terms here: 1. \(\sqrt{y-3}\), which can be rewritten as \((y-3)^{\frac{1}{2}}\) 2. \((y-3)^{5}\)

Step 2: Apply the product rule of exponentiation

Since both terms have the same base (y-3), we can apply the product rule of exponentiation. That means we need to add the exponents, which are \(\frac{1}{2}\) and 5 in this case. So, the product will be: \((y-3)^{\frac{1}{2} + 5}\)

Step 3: Simplify the exponent

To simplify the exponent, we can rewrite 5 as a fraction with a denominator of 2: \(5 = \frac{10}{2}\) Now, we can add the fractions: \(\frac{1}{2} + \frac{10}{2} = \frac{11}{2}\)

Step 4: Rewrite the expression with the simplified exponent

We can now rewrite our expression with the simplified exponent: \((y-3)^{\frac{11}{2}}\) So, the product of the given expressions is: \((y-3)^{\frac{11}{2}}\)

Key Concepts

Product Rule of Exponents

The product rule of exponents is a helpful tool to simplify expressions with the same base. When you multiply two exponents with the same base, you can simply add the exponents together. This works because multiplying powers of the same number is like repeated addition.
For example, in our exercise with the expressions \( \sqrt{y-3} \) and \((y-3)^5\), the base \((y-3)\) appears in both.
We rewrite \( \sqrt{y-3} \) as \((y-3)^{\frac{1}{2}}\) and keep \((y-3)^5\) as it is. Now, applying the product rule, we can combine them as \((y-3)^{\frac{1}{2} + 5}\). This makes the process straightforward, helping to manage complex algebraic expressions.

• Simplifies multiplying expressions with the same base
• Adds the exponents directly
• Reduces complexity in algebra

Simplifying Exponents

Simplifying exponents involves rewriting the expression in its most compact form. Once we apply the product rule, we need to simplify the resulting exponent. This often means dealing with fractions.
In our example, we have the expression \((y-3)^{\frac{11}{2}}\). To get there, we first converted the integer 5 into a fraction: \(5 = \frac{10}{2}\). Adding like fractions is straightforward: \(\frac{1}{2} + \frac{10}{2} = \frac{11}{2}\).
This step ensures that our expression is as neat as possible. Having the exponents in a simplified fraction form makes future calculations easier, allowing students to handle and manipulate expressions confidently.

• Rewrites integer exponents as fractions
• Adds fractions seamlessly
• Provides a clear, workable expression

Radical Expressions

Radical expressions involve roots, like square roots. They often need transformation into fractional exponents for simplification. This is key in our example where we had \(\sqrt{y-3}\).
The square root symbol \(\sqrt{}\) can be expressed as an exponent of \(\frac{1}{2}\). So, \(\sqrt{y-3}\) became \((y-3)^{\frac{1}{2}}\). Understanding this switch is vital for simplifying and manipulating algebraic expressions.
This conversion helps integrate radical expressions seamlessly into other algebraic operations, especially when applying rules like the product rule of exponents.

• Converts roots into fractional exponents
• Simplifies integration with polynomial expressions
• Facilitates algebraic manipulation