Problem 73 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

(a) \(4^{\frac{7}{3}}\), (b) \(3^3\), (c) \(5^{4k-2}\)."

Step 1: Simplify Expression (a)

We start by using the property of exponents that states: \(a^m \times a^n = a^{m+n}\). For expression (a), we have: \(4^{3} 4^{-\frac{2}{3}} = 4^{3 + (-\frac{2}{3})}\). Calculate the new exponent: \(3 + (-\frac{2}{3}) = 3 - \frac{2}{3} = \frac{9}{3} - \frac{2}{3} = \frac{7}{3}\).So, the simplified expression is \(4^{\frac{7}{3}}\).

Step 2: Simplify Expression (b)

We handle the exponents separately for multiplication and division. For multiplication in the numerator, use \(a^m \times a^n = a^{m+n}\):\((3^{2} \times 3^{\frac{1}{2}}) = 3^{2 + \frac{1}{2}} = 3^{2.5}\).For the division in the entire expression, use \(\frac{a^m}{a^n} = a^{m-n}\):\(\frac{3^{2.5}}{3^{-\frac{1}{2}}} = 3^{2.5 - (-\frac{1}{2})} = 3^{2.5 + \frac{1}{2}} = 3^{3}\).

Step 3: Simplify Expression (c)

Again, use the exponent rules for both multiplication and division. First, handle the multiplication in the numerator:\(5^k \times 5^{2k-1} = 5^{k + (2k-1)} = 5^{3k-1}\).For the entire division, use the formula \(\frac{a^m}{a^n} = a^{m-n}\):\(\frac{5^{3k-1}}{5^{1-k}} = 5^{(3k-1)-(1-k)} = 5^{3k-1-1+k} = 5^{4k-2}\).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is essential when working with exponential expressions. These properties simplify operations and make calculations much easier. Here are some key properties to remember:

• Product of Powers: The rule \(a^m \times a^n = a^{m+n}\) helps simplify multiplying expressions with the same base. By adding the exponents, this property makes the calculation straightforward.
• Power of a Power: To tackle \((a^m)^n\), you can use \(a^{m \times n}\). This involves multiplying the exponents, allowing you to condense the expression.
• Quotient of Powers: The rule \(\frac{a^m}{a^n} = a^{m-n}\) simplifies division by subtracting the exponent of the denominator from the exponent of the numerator.

These rules can simplify complex-looking problems and are the foundation of dealing with exponential expressions effectively. Consider them as your toolkit for mastering exponents!

Simplifying Expressions

Simplifying expressions involves rewriting them in an easier-to-manage form while maintaining their value. This process is essential when dealing with expressions containing exponents.
The strategy is to apply exponent properties to consolidate the expression. For instance, in the step for expression (a) \(4^3 \times 4^{-\frac{2}{3}}\), combining the exponents using the product of powers rule consolidates it to \(4^{\frac{7}{3}}\).
For expression (b), simplifying the division expression \(\frac{3^{2.5}}{3^{-\frac{1}{2}}}\) using the quotient of powers rule gives \(3^3\). By applying these steps, you transform complex expressions into simpler ones that are more comprehensible and manageable.

Power of a Quotient Rule

The power of a quotient rule is a vital concept when dealing with division involving exponents. It is stated as:

• \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\)

This rule helps you manage expressions where a power is applied to a fraction. Instead of taking each base to the power separately, it ensures the entire fraction is handled collectively, making calculations less error-prone.
In exercises like expression (b), \(\frac{3^{2} \times 3^{\frac{1}{2}}}{3^{-\frac{1}{2}}}\), breaking it down using this rule allows you to simplify it to \(3^3\) smoothly. Understanding this concept aids in simplifying operations where fractions and exponents intersect.

Power of a Product Rule

The power of a product rule is essential for handling expressions where an exponent applies to a product of factors. This rule is expressed as:

• \((ab)^n = a^n \times b^n\)

When an exponent affects multiple bases within a product, this rule distributes the exponent to each factor, simplifying algebraic manipulation. For example, if you encounter \((2 \times 3)^4\), you apply the rule to get \(2^4 \times 3^4\).
In the context of expression (c) with exponent multiplication occurrences like \(5^k \times 5^{2k-1}\), the power of a product isn't directly used, but understanding similar concepts helps in resolving compound expressions by distributing or consolidating exponents. This rule supports breaking down complex expressions into simpler factors, making problem-solving smoother.