Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Carry out the indicated operation and write your answer using positive exponents only. $$ \frac{4^{1 / 2}}{4^{5 / 2}} $$

Step-by-Step Solution

Answer

\(\frac{1}{16}\)

Step 1: Apply the quotient rule for exponents

Using the quotient rule, we will subtract the exponent in the denominator from the exponent in the numerator: \[ \frac{4^{1/2}}{4^{5/2}} = 4 ^{(1/2 - 5/2)} \]

Step 2: Subtract the exponents

Now we will subtract the exponents: \[ 4^{(1/2 - 5/2)} = 4^{- 4/2} \]

Step 3: Write the answer with positive exponents

To write the answer using positive exponents only, we'll use the exponent rule that states: \(a^{-n} = \frac{1}{a^n}\). So, we have: \[ 4^{- 4/2} = \frac{1}{4^{4/2}} \]

Step 4: Simplify the exponent

Now let's simplify the exponent in the denominator: \[ \frac{1}{4^{4/2}} = \frac{1}{4^2} \]

Step 5: Calculate the final result

Now calculate the result: \[ \frac{1}{4^2} = \frac{1}{16} \] Therefore, the answer is \(\frac{1}{16}\).

Key Concepts

Exponents

Exponents can be thought of as a shortcut for repeated multiplication. For example, when you see something like \(3^4\), it means \(3\) multiplied by itself four times: \(3 \times 3 \times 3 \times 3\). This is incredibly useful for handling large numbers or calculations where numbers repeat often.

The base of the exponent is the number that gets multiplied, and the exponent itself is the small number written above and to the right of the base, which tells you how many times to multiply the base by itself. For instance, in \(4^{1/2}\), \(4\) is the base, and \(1/2\) is the exponent.

• If the exponent is positive, like \(2^3\), it simply means multiply the base (\(2\)) three times: \(2 \times 2 \times 2 = 8\).
• If the exponent is zero, any base raised to the power of zero, except zero itself, is \(1\).
• If the exponent is negative, it indicates that you should divide by the base that many times instead of multiplying. This is why \(a^{-n} = \frac{1}{a^n}\).

Quotient Rule

The quotient rule for exponents is very handy when you are dividing two numbers (or algebraic terms) that have the same base. The rule states:

\(\frac{a^m}{a^n} = a^{m-n}\). This means you subtract the exponent in the denominator from the exponent in the numerator. So, if you have a fraction \(\frac{b^6}{b^2}\), you perform the exponent subtraction:

• \(6 - 2\), which results in \(b^{4}\)

Using this rule can make simplifying fractions with exponents much easier and more systematic, as it reduces the fraction to a single term with an exponent. Just make sure the base of both the numerator and denominator is the same when applying this rule.

Positive Exponents

Converting negative exponents to positive exponents is an essential skill in algebra. Negative exponents might seem a bit confusing at first but are simple once you grasp the concept.
According to the rule, \(a^{-n} = \frac{1}{a^n}\). This means if you have a negative exponent, like \(4^{-2}\), it would be expressed with a positive exponent by flipping it into a fraction:

• \(\frac{1}{4^2}\)

This conversion is crucial because solutions are often required with positive exponents, making them easier to understand and interpret. So, when you encounter terms with negative exponents, remember they can be rewritten as fractions with positive exponents.

Simplifying Expressions

Simplifying expressions is the process of making an expression more manageable and easier to use in calculations or further algebraic operations. It involves breaking down complex expressions into simpler forms. This is achieved by applying various rules and operations, such as the quotient rule for exponents or converting negative exponents to positives. Let’s take a closer look at how we simplified the original exercise expression:

• Started with \(\frac{4^{1/2}}{4^{5/2}}\).
• Applied the quotient rule, resulting in \(4^{(1/2 - 5/2)}\).
• Simplified to \(4^{-4/2}\), then used the positive exponent rule to achieve positive exponents: \(\frac{1}{4^{4/2}}\).
• Simplified further to \(\frac{1}{4^2}\), providing a more straightforward expression.

This process helps ensure that expressions are more straightforward for further mathematical operations or analyses.