Problem 73 - Calculus for Biology and Medicine

Question

Evaluate the following exponential expressions: (a) \(2^{4} 8^{-2 / 3}\) (b) \(\frac{3^{3} 3^{-1 / 2}}{3^{1 / 2}}\) (c) \(\frac{5^{k}(25)^{k-1}}{5^{2-k}}\)

Step-by-Step Solution

Answer

(a) 4, (b) 9, (c) 5^{4k-4}

Step 1: Simplify the First Expression

To solve part (a), we begin by simplifying each term separately: 1. Simplify the first term: The expression is \(2^4 = 16\).2. Simplify the second term: The expression is \(8^{-2/3} = (2^3)^{-2/3} = 2^{-2} = \frac{1}{4}\).Combine the results: \(16 \times \frac{1}{4} = 4\).

Step 2: Evaluate and Simplify Second Expression

For part (b), manage the terms with base 3:1. Simplify numerator: \(3^3 \times 3^{-1/2} = 3^{3 - 1/2} = 3^{5/2}\).2. The denominator is \(3^{1/2}\).Divide the simplified terms: \(\frac{3^{5/2}}{3^{1/2}} = 3^{(5/2)-(1/2)} = 3^{4/2} = 3^2 = 9\).

Step 3: Simplify the Third Expression

In part (c), simplify each factor using properties of exponents:1. Rewrite 25 as a power: \(25 = 5^2\), so \((25)^{k-1} = (5^2)^{k-1} = 5^{2(k-1)} = 5^{2k-2}\).2. Simplify the numerator: Combine terms: \(5^k \times 5^{2k-2} = 5^{k + 2k - 2} = 5^{3k - 2}\).3. Simplify the denominator: \(5^{2-k}\).4. Divide: \(\frac{5^{3k-2}}{5^{2-k}} = 5^{(3k-2)-(2-k)} = 5^{3k-2-2+k} = 5^{4k-4}\).The simplified expression: \(5^{4k-4}\).

Key Concepts

Properties of Exponents

The properties of exponents are fundamental tools that help us manipulate and simplify expressions that involve powers. When dealing with exponential expressions, such as \(2^n\) or \(a^b\), understanding these properties can make complex calculations much more manageable. Here are some key properties that are particularly useful:

• Product of Powers: When multiplying two expressions with the same base, you add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When raising an exponent to another power, you multiply the exponents. So, \((a^m)^n = a^{m \times n}\).
• Power of a Product: When you take a power of a product, the exponent is distributed to each factor of the product. Hence, \((ab)^n = a^n \times b^n\).
• Quotient of Powers: This property applies when you divide two expressions with the same base. You subtract the exponent of the denominator from the exponent of the numerator: \(\frac{a^m}{a^n} = a^{m-n}\).

Using these properties appropriately not only simplifies expressions but also reveals hidden relations between variables, aiding in error reduction during calculations.

Expression Simplification

Expression simplification involves breaking down complex mathematical expressions into simpler forms, which are easier to understand and solve. Let's revisit parts of the original exercise: simplifying expressions that include different operations such as multiplication, division, and applying powers simultaneously.For instance, in the expression \(\frac{3^3 \times 3^{-1/2}}{3^{1/2}}\), each term in the expression has the same base, 3. To simplify it:

• In the numerator, you use the product of powers property: \((3^3 \times 3^{-1/2}) = 3^{3 - 1/2} = 3^{5/2}\).
• For the entire expression, apply the quotient of powers property: \(\frac{3^{5/2}}{3^{1/2}} = 3^{5/2 - 1/2} = 3^{4/2} = 3^2 = 9\).

This step-by-step simplification reveals that the original expression can be rewritten as a much more manageable integer, illustrating how exponent properties help streamline the process.

Exponent Rules

Exponent rules are like the building blocks that allow you to systematically work your way through any problem involving powers. They dictate how exponents interact with each other in expressions.Some fundamental rules include:

• Zero Exponent Rule: Any non-zero number raised to the power of zero is 1, such as \(a^0 = 1\).
• Negative Exponent Rule: A negative exponent means that the base is on the wrong side of a fraction line, or it can be flipped across the fraction line to make it positive: \(a^{-n} = \frac{1}{a^n}\).

Applying these rules can make simplifying expressions more intuitive. Consider the expression \(8^{-2/3}\). Recognizing that 8 can be rewritten as \(2^3\), and employing the negative exponent rule helps you to express it as \(\frac{1}{4}\). Consistently using exponent rules not only clarifies expressions but also fosters a deeper understanding of exponential relationships, enabling you to solve complex problems with greater ease and accuracy.