Problem 15 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without using exponents. $$ \frac{2^{3} \cdot 2^{5}}{2^{4} \cdot 2^{9}} $$

Step-by-Step Solution

Answer

The simplified expression without using exponents is \(\frac{1}{32}\).

Step 1: Use the exponent rule for multiplication

Using the exponent rule for multiplying exponents with the same base, we can add the exponents, which is given as: \(a^{m} \cdot a^{n} = a^{m+n}\) For the expression \(\frac{2^{3} \cdot 2^{5}}{2^{4} \cdot 2^{9}}\), we will first add the exponents in the numerator and the exponents in the denominator separately: \[ \frac{2^{(3+5)}}{2^{(4+9)}}\]

Step 2: Simplify the expression

Now, we can simplify the expression by doing the arithmetic operations: \[ \frac{2^{8}}{2^{13}}\]

Step 3: Use the exponent rule for division

Use the exponent rule for dividing exponents with the same base, which is given as: \[ \frac{a^{m}}{a^{n}} = a^{m-n}\] Applying this rule to the expression, we get: \[ 2^{8-13}\]

Step 4: Simplify the expression

Simplify the expression by subtracting the exponents: \[ 2^{-5}\] The final expression without using exponents is \(2^{-5}\), which is an acceptable way to present the simplified expression. However, since the student is asked to rewrite the number without using exponents, we will continue to the next step.

Step 5: Convert negative exponent to positive exponent

Since \(2^{-5}\) has a negative exponent, we can convert it to a positive exponent by taking its reciprocal: \[ 2^{-5} = \frac{1}{2^{5}}\]

Step 6: Compute the value

Finally, compute the value of the expression: \[ \frac{1}{2^{5}} = \frac{1}{32}\] Thus, the number \(\frac{2^{3} \cdot 2^{5}}{2^{4} \cdot 2^{9}}\) without using exponents is \(\frac{1}{32}\).

Key Concepts

Multiplication of Exponents

When multiplying exponents with the same base, you can simplify the process by adding the exponents together. This is one of the basic rules of exponents that helps in reducing expressions neatly. For instance, if we have the expression \(a^{m} \cdot a^{n}\), we should add the exponents \(m\) and \(n\) since the bases are the same. Thus, it becomes \(a^{m+n}\).

This concept was applied in the problem to merge the terms in the numerator: \(2^{3} \cdot 2^{5}\) became \(2^{8}\) by simply adding the exponents 3 and 5. Recognizing when bases are the same and applying this rule helps in tackling more complicated expressions efficiently.
Remember, there are a few exceptions, always ensure that the bases are exactly the same before applying these addition rules.

Division of Exponents

In exponentiation, division involves subtracting exponents when the base remains unchanged. This technique helps simplify complex fractions that have powers in both the numerator and the denominator. The rule for division of exponents with the same base is given by \(\frac{a^{m}}{a^{n}} = a^{m-n}\).

In the provided solution, we used this rule to simplify \(\frac{2^{8}}{2^{13}}\). By subtracting the exponents, 13 from 8, we found \(2^{-5}\).
This subtraction of exponents helps greatly in reducing the complexity of the expression, allowing us to express large fractional powers more concisely. Always ensure that the bases are identical when using this rule to avoid errors.

Negative Exponents

A negative exponent signifies the reciprocal of the base raised to the positive exponent. For example, \(a^{-n}\) is equivalent to \(\frac{1}{a^{n}}\). This is a handy way of converting an expression with a negative exponent into its positive form, making computations more straightforward.

In simplifying \(2^{-5}\), recognizing it as \(\frac{1}{2^{5}}\) changes a potentially confusing term into one that's much easier to compute. The expression \(\frac{1}{32}\) follows directly once you recognize the power of 2 that 32 represents.
Using this rule transforms difficult expressions into more intelligible forms, crucial for accurate simplification and computation of exponents.