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^{8.5}, (c) 3^{-9-9k}

Step 1: Simplify Inside the Parentheses for Expression (a)

First, let's look at the expression \( \left(2^{4} 2^{-3/2}\right)^{2} \). Apply the product of powers rule, which states that when multiplying like bases, you add the exponents:\[2^{4} 2^{-3/2} = 2^{4 + (-3/2)} = 2^{5/2}\].The expression becomes \( \left(2^{5/2}\right)^{2} \).

Step 2: Apply the Power of a Power Rule for Expression (a)

Now the expression is \( \left(2^{5/2}\right)^{2} \). Use the power of a power rule, which says you multiply the exponents:\[\left(2^{5/2}\right)^{2} = 2^{(5/2) \times 2} = 2^{5}\].Thus, \( 2^{5} = 32 \).

Step 3: Simplify Inside the Parentheses for Expression (b)

Consider the expression \( \left(\frac{6^{5/2} 6^{2/3}}{6^{1/3}}\right)^{3} \).First, simplify inside the parentheses by combining the numerators, applying the product of powers rule:\[6^{5/2} \cdot 6^{2/3} = 6^{5/2 + 2/3}\].To combine the exponents, find a common denominator to get:\[5/2 = 15/6 \quad \text{and} \quad 2/3 = 4/6,\]so we have:\[6^{15/6 + 4/6} = 6^{19/6}\].Now divide by \( 6^{1/3} \), or equivalently, subtract the exponents:\[6^{19/6 - 1/3} = 6^{19/6 - 2/6} = 6^{17/6}\].Thus, the simplified expression is \( \left(6^{17/6}\right)^{3} \).

Step 4: Apply the Power of a Power Rule for Expression (b)

Now it's \( \left(6^{17/6}\right)^{3} \). Apply the power of a power rule:\[\left(6^{17/6}\right)^{3} = 6^{(17/6) \times 3} = 6^{51/6}\].Simplifying \(51/6\) gives \(6^{8.5}\).

Step 5: Simplify Inside the Parentheses for Expression (c)

For expression (c), \( \left(\frac{3^{-2k+3}}{3^{4+k}}\right)^{3} \), start by simplifying inside the parentheses:Divide the exponents by subtracting:\[\frac{3^{-2k+3}}{3^{4+k}} = 3^{-2k+3-(4+k)} = 3^{-2k+3-4-k} = 3^{-3-3k}\].Now you have \( \left(3^{-3-3k}\right)^{3} \).

Step 6: Apply the Power of a Power Rule for Expression (c)

Now apply the power of a power rule to \( \left(3^{-3-3k}\right)^{3} \):\[\left(3^{-3-3k}\right)^{3} = 3^{(-3-3k) \times 3} = 3^{-9-9k}\].

Key Concepts

Power of a Power Rule

The power of a power rule is crucial in dealing with nested exponents. It tells us that when you raise a power to another power, you should multiply the exponents. For instance, if you have \(a^{m}\) raised to the power of \(n\), it can be rewritten simply as \(a^{m \times n}\).

Apply this rule whenever you come across something like \(\left(a^{m}\right)^{n}\). Let's look at an example where \(2^{5/2}\) is squared. By multiplying \(5/2 \) and \(2\), we get \(2^{5}\). This means every base of 2 in the expression was used twice in the powering processes, hence the final \(2^{5} = 32\).

Understanding this rule helps in tackling more complex exponential expressions efficiently, reducing them to a more manageable form.

Product of Powers Rule

The product of powers rule is used to simplify expressions where you multiply like bases. It states that you add the exponents when the bases are the same, transforming the process into simple addition of the exponents. For instance, \(a^m \times a^n = a^{m+n}\).

This rule works beautifully in example (a), where the bases \(2\) with exponents \(4\) and \(-3/2\) were combined. Doing the arithmetic, \(2^{4} \times 2^{-3/2}\) simplifies to \(2^{5/2}\). Similarly, in example (b), you sum up \(5/2\) and \(2/3\) for the base of 6 to get \frac{19}{6}\. Counting in terms of a common denominator makes it simple and straightforward.

By applying this rule, not only does the expression become simpler, but you also lay a strong foundation for more advanced math.

Simplifying Exponents

Simplifying exponents is about turning complex expressions into simpler ones by using exponent rules effectively. The aim is to reduce clutter by cleverly adding, subtracting, or multiplying exponents as required.

It's all about finding ways to express your result as a single power, ideally by having one fewer operation than you started with. For instance, we encounter \(\frac{6^{19/6}}{6^{1/3}}\) simplifying down to \(6^{17/6}\) using exponent subtraction. The same goes for expression \(c\), where complex bases and exponents are simplified thoroughly before extending further with power of a power rule.

This valuable skill smoothens your journey through algebra and prepares you for calculus. It teaches one to bring things under control methodically by seeing through the workload of exponents.

Algebraic Manipulation

Algebraic manipulation is the art of rearranging equations or expressions to simplify them or solve for a variable. This involves performing valid operations to uncover hidden solutions or make equations more interpretable.

This process calls for familiarity with mathematical properties and laws. In our examples, algebraic manipulation bridges different stages of simplification. When you have to deal with something like \(3^{-2k+3-4-k}\), it's all about collecting like terms, performing arithmetic operations, and then slicing through complexity by framing clear, power-elegant conclusions. The task often appears in sequences of simplifying and combining terms accurately before performing exponent saving tricks.

Through mastery of manipulation methods, solving expressions and equations becomes less of a mystery and more of a systematic procedure.