Problem 99 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \sqrt{b} \cdot \sqrt[4]{b} $$

Step-by-Step Solution

Answer

The expression simplifies to \( b^{\frac{3}{4}} \).

Step 1: Express roots as exponents

Convert the roots into expressions with rational exponents. The square root of any number can be expressed as an exponent of \( \frac{1}{2} \). So, \( \sqrt{b} = b^{\frac{1}{2}} \). The fourth root can be expressed as an exponent of \( \frac{1}{4} \), so \( \sqrt[4]{b} = b^{\frac{1}{4}} \).

Step 2: Multiply expressions with the same base

When you multiply expressions with the same base, you add their exponents. Write the product of \( b^{\frac{1}{2}} \) and \( b^{\frac{1}{4}} \) as \( b^{\frac{1}{2} + \frac{1}{4}} \).

Step 3: Add the exponents

To add the exponents \( \frac{1}{2} \) and \( \frac{1}{4} \), find a common denominator. The common denominator of 2 and 4 is 4. Convert \( \frac{1}{2} \) to \( \frac{2}{4} \). So, \( \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \).

Step 4: Write the simplified expression

Now that you've determined the sum of the exponents, write the simplified expression as \( b^{\frac{3}{4}} \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves rewriting an expression in a more manageable or understandable form without changing its value. In the case of the given expression \( \sqrt{b} \cdot \sqrt[4]{b} \), simplifying means expressing the product in a form that is easier to understand or calculate.
To start, it's crucial to convert any roots into rational exponents. This simplifies multiplication or division, making them much easier to handle. For instance:

• The square root \( \sqrt{b} \) becomes \( b^{\frac{1}{2}} \).
• The fourth root \( \sqrt[4]{b} \) becomes \( b^{\frac{1}{4}} \).

Rational exponents allow us to apply simpler arithmetic operations by working with fractions. Additionally, converting roots into exponents sets the stage for multiplying terms with the same base, turning complex operations into straightforward arithmetic.
Once roots are converted into rational exponents, simplifying the expression involves making use of exponent rules, allowing us to consolidate the expression neatly.

Multiplying Exponents

Multiplying exponents involves a rule specific to expressions with the same base.
When multiplying two powers that have the same base, you add the exponents together. This rule relies on the fundamental understanding that multiplying powers is akin to combining the repeated multiplication of the base:
For example, consider the expression \( b^{\frac{1}{2}} \cdot b^{\frac{1}{4}} \).

• The bases (\(b\)) are the same.
• Add the exponents: \( \frac{1}{2} + \frac{1}{4} \).

The result of this addition leads to a single, simplified power: \( b^{\frac{3}{4}} \). Adding the exponents facilitates the process of reducing expressions, leaving you with a cleaner, more effective expression.
This method not only simplifies calculations but also reduces risks of errors, especially in complex problems where matching bases occur frequently.

Exponent Rules

Exponent rules are essential guidelines that help simplify mathematical operations involving powers. They help reduce what might seem like daunting arithmetic into straightforward steps.
Here's a quick refresher on some critical exponent rules:

• Product of Powers Rule: When multiplying two expressions with the same base, add the exponents. For example, \( a^m \cdot a^n = a^{m+n} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents. For instance, \( (a^m)^n = a^{m\cdot n} \).
• Power of a Product Rule: When a product is raised to a power, apply the exponent to all factors in the product: \( (ab)^m = a^m \cdot b^m \).
• Negative Exponent Rule: An expression with a negative exponent can be transformed into its reciprocal raised to the positive exponent: \( a^{-n} = \frac{1}{a^n} \).

Understanding and applying these rules allows one to skillfully manipulate and simplify any expressions, making complex equations much more approachable and easy to solve.
Remember, all these rules work best when the base is the same across the terms you're working with, making them highly useful for simplifying expressions like in the original exercise.