Problem 70 - Algebra and Trigonometry

Question

\(69-82\) . Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. $$ \sqrt[4]{b^{3}} \sqrt{b} $$

Step-by-Step Solution

Answer

The simplified expression is \\(b^{\frac{5}{4}}\\).

Step 1: Convert Radicals to Exponents

First, let's convert each radical expression into one that uses exponents. Recall that the square root of a number is the same as raising it to the power of \(\frac{1}{2}\), and the fourth root is equivalent to raising the number to the power of \(\frac{1}{4}\). Thus, \sqrt[4]{b^3}\ becomes \(b^{\frac{3}{4}}\) and \sqrt{b}\ becomes \(b^{\frac{1}{2}}\).

Step 2: Multiply the Expressions Using Exponents

Next, we multiply the expressions \(b^{\frac{3}{4}}\) and \(b^{\frac{1}{2}}\) together. When multiplying exponents with the same base, we add the exponents: \(b^{\frac{3}{4} + \frac{1}{2}}\).

Step 3: Find a Common Denominator and Simplify

To add \(\frac{3}{4}\) and \(\frac{1}{2}\), we must have a common denominator. The common denominator for 4 and 2 is 4. Convert \(\frac{1}{2}\) to \(\frac{2}{4}\). Now add the exponents: \(\frac{3}{4} + \frac{2}{4} = \frac{5}{4}\).

Step 4: Express the Final Result

The expression \(b^{\frac{3}{4}} \cdot b^{\frac{1}{2}}\) simplifies to \(b^{\frac{5}{4}}\). This uses rational exponents to express the original expression.

Key Concepts

Understanding Radicals

Radicals are expressions that involve roots, such as square roots or cube roots. They are a fundamental part of algebra and are often used to simplify complex expressions. For example, the square root of a number is the number that, when multiplied by itself, gives the original number. The primary symbol for the square root is \( \sqrt{} \), followed by the number or expression under the root.

Different types of roots include:

• Square root (\( \sqrt{x} \)): the root of a degree of 2
• Cubic root (\( \sqrt[3]{x} \)): the root of a degree of 3
• Fourth root (\( \sqrt[4]{x} \)): the root of a degree of 4

Radicals can be converted into exponent notation. For instance, \( \sqrt[4]{b^3} \) can be expressed as \( b^{3/4} \), showing that the expression is raised to a fractional power, where the numerator is the power of the base and the denominator is the type of root. This conversion helps in the application of exponent rules.

Exploring Exponent Rules

Exponent rules are crucial when working with expressions involving powers. They allow us to simplify expressions and understand how different terms interact. Let's break down some of the fundamental rules for exponents:

• Product Rule: When you multiply two exponential expressions with the same base, add their exponents: \( a^m \cdot a^n = a^{m+n} \).
• Quotient Rule: Dividing two expressions with the same base means subtracting exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power Rule: When raising an exponent to another power, multiply the exponents: \( (a^m)^n = a^{mn} \).
• Negative Exponent Rule: A negative exponent indicates the reciprocal of the base: \( a^{-m} = \frac{1}{a^m} \).

In the context of the original exercise, we use the product rule. Converting radical expressions to exponents allows for easy multiplication of terms with identical bases. Adding exponents together is a straightforward process once they are in similar fractional forms.

Simplifying Expressions with Rational Exponents

Simplifying algebraic expressions becomes intuitive when using rational exponents. This form of notation offers a powerful way to combine radicals and exponents into a single manageable expression. Let's explore simplifying the expression from the exercise:

When we initially convert radicals to exponents, we transform an otherwise complex radical expression into a more straightforward power expression. This makes it possible to leverage exponent rules to simplify effectively. For example, to simplify \( \sqrt[4]{b^{3}} \sqrt{b} \):

• Convert the radicals: \( \sqrt[4]{b^3} = b^{3/4} \) and \( \sqrt{b} = b^{1/2} \)
• Apply the product rule for exponents by adding their powers: \( b^{3/4 + 1/2} \)
• Find a common denominator to simplify addition: \( 3/4 + 2/4 = 5/4 \)
• The simplified form is \( b^{5/4} \) using rational exponents.

This process showcases both the method and flexibility of using rational exponents, allowing for streamlined operations without directly dealing with roots during calculations.