Problem 74 - Calculus for Biology and Medicine

Question

Evaluate the following exponential expressions: (a) \(\left(2^{4} 2^{-3 / 2}\right)^{2}\) (b) \(\left(\frac{6^{5 / 2} 6^{2 / 3}}{6^{1 / 3}}\right)^{3}\) (c) \(\left(\frac{3^{-2 k+3}}{3^{4+k}}\right)^{3}\)

Step-by-Step Solution

Answer

(a) 32; (b) \(6^{17/2}\); (c) \(3^{-9k-3}\).

Step 1: Simplify the Base of Expression (a)

Start by simplifying the base inside the parenthesis: \(2^{4} 2^{-3/2}\) using the property \(a^{m} \cdot a^{n} = a^{m+n}\). \[2^{4} 2^{-3/2} = 2^{4 - 3/2} = 2^{5/2}\]

Step 2: Raise to the Power for Expression (a)

Raise the simplified base to the power of 2: \[\left(2^{5/2}\right)^{2} = 2^{(5/2) \cdot 2} = 2^{5} = 32\]

Step 3: Simplify the Base of Expression (b)

Simplify the base inside the parenthesis for expression (b): \[\frac{6^{5/2} \cdot 6^{2/3}}{6^{1/3}}\] Start by combining the powers in the numerator using \(a^{m} \cdot a^{n} = a^{m+n}\): \[6^{5/2 + 2/3} = 6^{15/6 + 4/6} = 6^{19/6}\]Then simplify the fraction using \(\frac{a^{m}}{a^{n}} = a^{m-n}\): \[6^{19/6 - 1/3} = 6^{19/6 - 2/6} = 6^{17/6}\]

Step 4: Raise to the Power for Expression (b)

Raise the simplified base to the power of 3:\[\left(6^{17/6}\right)^{3} = 6^{(17/6) \cdot 3} = 6^{17/2}\]

Step 5: Simplify the Base for Expression (c)

Simplify the base of expression (c): \[\frac{3^{-2k+3}}{3^{4+k}}\] Using the property \(\frac{a^{m}}{a^{n}} = a^{m-n}\), simplify the expression to:\[3^{-2k+3-(4+k)} = 3^{-2k+3-4-k} = 3^{-3k-1}\]

Step 6: Raise to the Power for Expression (c)

Raise the simplified base to the power of 3: \[\left(3^{-3k-1}\right)^{3} = 3^{(-3k-1) \cdot 3} = 3^{-9k-3}\]

Key Concepts

Exponent Rules

Exponent rules are like guidelines for handling the powers of numbers. When you multiply numbers with the same base, you can simply add their exponents. This is captured in the formula:

• \(a^m \cdot a^n = a^{m+n}\)

These rules help streamline calculations and are essential when working with complex expressions.
Besides multiplying, you also have rules for dividing powers. When you divide, subtract the exponents:

• \(\frac{a^m}{a^n} = a^{m-n}\)

These rules allow for simplifying otherwise complicated expressions, making them easier to manage.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. Let's take a deeper dive into this process.
For instance, consider the expression \(\left(2^4 \cdot 2^{-3/2}\right)^2\). First, you combine the exponents inside the parenthesis using the rules for multiplication of powers:

• \(2^{4 - 3/2} = 2^{5/2}\)

Then, raise the simplified base to the power outside the parenthesis. This transforms it into a simpler expression:

• \(\left(2^{5/2}\right)^2 = 2^5 = 32\)

This process can involve combining terms, cancelling out numbers, or performing additional arithmetic. Simplification is a crucial step in solving any algebraic or exponential equation.

Power of a Power Property

The power of a power property is an essential tool that lets you simplify expressions where an exponent is raised to another exponent. This property is defined by:

• \(\left(a^m\right)^n = a^{m \cdot n}\)

It essentially means that when you see an expression like \(\left(a^m\right)^n\), you multiply the exponents together.
Applying this to of one our examples:

• \(\left(6^{17/6}\right)^3 = 6^{(17/6) \cdot 3} = 6^{17/2}\)

These steps not only help simplify complex expressions but also make it easier to evaluate and solve them. Mastery of this property is key to tackling exponential equations effectively.