Problem 68 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by band. Approximate the answer to the nearest hundredth when appropriate. $$ \frac{6^{1 / 5} \cdot 6^{3 / 5}}{6^{2 / 5}} $$

Step-by-Step Solution

Answer

The expression evaluates to approximately 2.05.

Step 1: Simplify the Expression Using Laws of Exponents

First, recognize that the expression involves multiplying and dividing powers with the same base, which is 6 in this case. According to the law of exponents, for any real numbers \( a \) and any integers \( m \) and \( n \), \( a^m \cdot a^n = a^{m+n} \) and \( \frac{a^m}{a^n} = a^{m-n} \). Apply these rules to simplify the given expression:\[\frac{6^{1/5} \cdot 6^{3/5}}{6^{2/5}} = \frac{6^{(1/5 + 3/5)}}{6^{2/5}} = \frac{6^{4/5}}{6^{2/5}}\]

Step 2: Further Simplify the Exponent

Now, continue simplifying using the quotient rule \( \frac{a^m}{a^n} = a^{m-n} \):\[6^{4/5 - 2/5} = 6^{2/5}\]

Step 3: Calculate the Result

Now calculate the value of \( 6^{2/5} \) to approximate the answer to the nearest hundredth. This involves finding the fifth root of 6 squared:\[6^{2/5} = (6^{1/5})^2 \]Calculate \( 6^{1/5} \), which is approximately 1.43097. Thus,\[6^{2/5} \approx 1.43097^2 \approx 2.04767\]Rounding to the nearest hundredth gives:\[6^{2/5} \approx 2.05\]

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making mathematical statements more manageable by applying rules and patterns. When working with exponents, one of the key strategies is to use the "Laws of Exponents." These laws help us combine or reduce expressions that involve powers. In this particular exercise, we deal with a fraction containing powers of 6. First, note that when the same base is involved in exponents, you can use the following rules:

• Multiplying Exponents: \(a^m \cdot a^n = a^{m+n}\)
• Dividing Exponents: \(\frac{a^m}{a^n} = a^{m-n}\)

By utilizing these rules, we can simplify the original expression \(\frac{6^{1/5} \cdot 6^{3/5}}{6^{2/5}}\) to just a single power: \(6^{2/5}\). This process helps us focus on a simpler form, which is much easier to handle for further calculations and approximations.

Quotient Rule

The quotient rule is a powerful tool in the realm of exponents that deals with division. If you have a situation where you're dividing like bases raised to powers, the rule says: "subtract the exponents." With a statement like \(\frac{a^m}{a^n} = a^{m-n}\), you're essentially reducing the expression to a manageable form.Let's see this in action. In the given problem, once we simplified the expression to \(\frac{6^{4/5}}{6^{2/5}}\), the next step was to apply the quotient rule. We subtract the exponent in the denominator (\(2/5\)) from the exponent in the numerator (\(4/5\)). This results in a new single power of \(6\) with an exponent of \(2/5\). This process doesn’t change the actual value but makes it easier to compute or approximate the result.

Approximations

Approximations in math aim to provide a practical estimate of a number's value, especially when dealing with irrational numbers or roots that may not be easily calculated in exact terms. In this exercise, after simplifying the expression to \(6^{2/5}\), we want to find a numerical estimate of this value.Calculating this involves understanding fractional exponents. The expression \(6^{2/5}\) is equivalent to squaring the fifth root of 6. To approximate \(6^{1/5}\), calculators or tables are often used, giving us about 1.43097. Then, we square this value: \(1.43097^2 \approx 2.04767\). For practical uses, we round to the nearest hundredth: \(2.05\). This step ensures our answer is precise enough for assignments while recognizable and understandable. By using approximations, we turn complex mathematical expressions into amounts we can easily interpret and use.