Problem 11 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. $$ 27^{5 / 6} \cdot 27^{-1 / 6} $$

Step-by-Step Solution

Answer

The expression evaluates to 9.

Step 1: Apply the Power Property of Exponents

Start by writing the expression in a form that allows you to use the power property, which states that when you multiply expressions with the same base, you add the exponents. The expression is \( 27^{5/6} \cdot 27^{-1/6} \).

Step 2: Combine the Exponents

Using the power property, combine the exponents: \( 27^{5/6 + (-1/6)} = 27^{(5/6 - 1/6)} \).

Step 3: Simplify the Exponent

Subtract \(-1/6\) from \(5/6\). This gives \(5/6 - 1/6 = 4/6\), which simplifies to \(2/3\). Therefore, you have \(27^{2/3}\).

Step 4: Express with a Radical and Simplify

Recognize that \(27^{2/3}\) can be rewritten in radical form as \((\sqrt[3]{27})^2\). First, find \(\sqrt[3]{27}\), which is 3 because \(3^3 = 27\). Then, calculate \(3^2 = 9\).

Key Concepts

Power Property of Exponents

In mathematics, the power property of exponents is a vital rule that helps simplify expressions with the same base. If you grasp this concept, you can effectively manage complex algebraic expressions.
When multiplying expressions with the same base, you add the exponents together. This property is expressed as follows:

• For any non-zero number a, and integers m and n, we have: \(a^m \cdot a^n = a^{m+n}\).

It's important because it allows you to simplify powers of the same base into a single exponent. For example, in the expression \(27^{5/6} \cdot 27^{-1/6}\), the base is the same (27), so you add the exponents: \(5/6 + (-1/6)\).
Once combined, the expression becomes easier to evaluate. This property is a basic building block that makes larger equations less intimidating.

Simplifying Exponents

Simplifying exponents involves reducing an expression to its simplest form. This often means combining, adding, or subtracting exponents when possible. To effectively simplify exponents, follow these general steps:

• Identify terms with the same base.
• Add or subtract the exponents according to the power property.
• Simplify any fractions in the resulting exponent.

In our exercise, we initially have the exponents \(5/6\) and \(-1/6\). Start by combining them using the power property, resulting in \(4/6\) after subtraction. This number is then simplified to \(2/3\).
This simplification is crucial as it reduces the complexity of solving the problem and leads to a more manageable expression, i.e., \(27^{2/3}\). Understanding this step is key to handling exponential equations confidently.

Radical Expressions

Radical expressions involve roots and can initially seem complex, but they follow systematic rules that make them easier to work with. Understanding how exponents relate to roots is essential.
First, a radical expression such as \(a^{m/n}\) can be interpreted as "the n-th root of a raised to the m." So, when we have \(27^{2/3}\), it implies:

• The cube root of 27 (\(\sqrt[3]{27}\)).
• This result, 3, is then squared (\(3^2 = 9\)).

Radicals are useful for expressing non-integer exponents in a familiar format. Converting between exponents and radicals helps simplify and solve equations, making the expression less daunting visually.
Once you translate the expression into radical form, such as \((\sqrt[3]{27})^2\), it reveals the straightforward calculation of reaching 9. Remember, mastering the movement between exponents and radicals will enhance your algebra skills significantly.