Problem 62 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive and write your answer in radical notation. $$ \sqrt[3]{a^{3} b^{2}} \cdot \sqrt{a^{2} b} $$

Step-by-Step Solution

Answer

The simplified expression is \( a^2 b \sqrt[6]{b} \).

Step 1: Rewrite using Exponents

First, rewrite each radical using exponents. A cube root can be written as an exponent of \( \frac{1}{3} \), and a square root can be written as an exponent of \( \frac{1}{2} \):\[ \sqrt[3]{a^3 b^2} = (a^3 b^2)^{\frac{1}{3}} \] and \[ \sqrt{a^2 b} = (a^2 b)^{\frac{1}{2}} \]

Step 2: Apply the Exponent Rule

Use the power of a product rule, which states \((xy)^m = x^m \, y^m\), to separate the terms inside each radical.\[ (a^3 b^2)^{\frac{1}{3}} = a^{3 \cdot \frac{1}{3}} \cdot b^{2 \cdot \frac{1}{3}} = a^1 \cdot b^{\frac{2}{3}} = a \cdot b^{\frac{2}{3}} \] and \[ (a^2 b)^{\frac{1}{2}} = a^{2 \cdot \frac{1}{2}} \cdot b^{1 \cdot \frac{1}{2}} = a^1 \cdot b^{\frac{1}{2}} = a \cdot b^{\frac{1}{2}} \]

Step 3: Multiply the Expressions

Now multiply the simplified expressions obtained from each radical. Combine \( a^1 \) terms and the \( b \) exponent terms.\[ a \cdot b^{\frac{2}{3}} \cdot a \cdot b^{\frac{1}{2}} = a^2 \cdot b^{\frac{2}{3} + \frac{1}{2}} = a^2 \cdot b^{\frac{4}{6} + \frac{3}{6}} = a^2 \cdot b^{\frac{7}{6}} \]

Step 4: Convert Back to Radical Notation

Finally, convert the fractional exponent back into radical notation for \( b \). The \( \frac{7}{6} \) exponent is a combination of a cube root and a sixth power, written as \( b^{\frac{1}{6} \cdot 7} \):\[ a^2 \cdot b^{\frac{7}{6}} = a^2 \cdot b^{1 + \frac{1}{6}} = a^2 \cdot b \cdot b^{\frac{1}{6}} = a^2 \cdot b \cdot \sqrt[6]{b} \]

Step 5: Final Solution

The simplified expression in radical notation is \( a^2 b \sqrt[6]{b} \).

Key Concepts

Radical Notation

Radical notation is a way of writing expressions that include roots, such as the square root and cube root. In mathematics, a root is essentially the opposite of an exponent. While exponents tell us how many times to multiply a number by itself, roots tell us what number can be multiplied by itself a given number of times to reach the original number. You might often see notations like \( \sqrt{x} \) or \( \sqrt[n]{x} \) being used.

Here's a simple breakdown:

• The symbol \( \sqrt{} \) represents the square root, where no index number is visible.
• When you see a number inside the root symbol, like \( \sqrt[3]{x} \), it indicates a cube root.
• The expression inside the root sign is called the radicand.

Using radical notation can simplify complex expressions into a format that's easier to manage when performing algebraic tasks. This is especially useful in calculations involving roots because it allows you to work with them in a systematic way by applying exponent rules.

Exponent Rules

Exponent rules are crucial for simplifying expressions involving powers. These rules help you manipulate and simplify expressions involving exponents in a clear and manageable manner.

Some key rules include:

• Power of a product: \( (xy)^m = x^m \cdot y^m \)
• Power of a power: \( (x^m)^n = x^{mn} \)
• Product of powers: \( x^m \cdot x^n = x^{m+n} \)
• Quotient of powers: \( \frac{x^m}{x^n} = x^{m-n} \)

In the solution to this problem, exponent rules help us break down radical notation by expressing roots as fractional exponents. For instance, the cube root \( \sqrt[3]{a^3 b^2} \) is expressed as \( (a^3 b^2)^{\frac{1}{3}} \). By applying these rules systematically, we simplify the given expressions before multiplying them.

Cube Root

A cube root is the inverse operation of cubing a number. When you take the cube root of a number, you are finding what number, when multiplied by itself three times, gives you the original number. It is expressed in radical notation as \( \sqrt[3]{x} \).

For example:

• \( \sqrt[3]{8} = 2 \), because \( 2 \cdot 2 \cdot 2 = 8 \).
• \( \sqrt[3]{27} = 3 \), since \( 3 \cdot 3 \cdot 3 = 27 \).

In algebra, cube roots can be rewritten using fractional exponents. This means \( \sqrt[3]{x} \) is equivalent to \( x^{\frac{1}{3}} \). This conversion is pivotal in simplifying algebraic expressions, as it allows you to use exponent rules effectively. For instance, in the expression \( \sqrt[3]{a^3 b^2} = (a^3 b^2)^{\frac{1}{3}} \), breaking it down further using exponent rules makes simplification possible.

Square Root

A square root is an operation that finds a number which, when multiplied by itself, equals the original number. Often used in various levels of mathematics, it's written in radical notation as \( \sqrt{x} \).

Here's how it works:

• \( \sqrt{9} = 3 \), as \( 3 \times 3 = 9 \).
• \( \sqrt{16} = 4 \), because \( 4 \times 4 = 16 \).

In algebra, expressing square roots with fractional exponents is common. This means \( \sqrt{x} = x^{\frac{1}{2}} \). Using this notation, we can apply the exponent rules, like in our exercise, which facilitates the simplification of complex expressions. Therefore, transforming a square root to \( (a^2 b)^{\frac{1}{2}} \) allows us to handle it using familiar algebraic methods and make it easier to combine and simplify with other terms.