Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Carry out the indicated operation and write your answer using positive exponents only. $$ \frac{2^{-1 / 2} \cdot 3^{2 / 3}}{2^{3 / 2} \cdot 3^{-1 / 3}} $$

Step-by-Step Solution

Answer

The simplified expression using positive exponents only is: \(\boxed{\frac{3}{4}}\).

Step 1: Rewrite the expression with brackets

To make it clear(er) which numerators and denominators belong to which bases, we rewrite the expression with brackets: \[ \frac{(2^{-1/2})(3^{2/3})}{(2^{3/2})(3^{-1/3})} \]

Step 2: Divide like bases with exponents

We're going to use the rule for dividing like bases (\(\frac{a^m}{a^n} = a^{m - n}\)): \[ \begin{aligned} \frac{2^{-1/2}}{2^{3/2}} &= 2^{-1/2 - 3/2} \\ \frac{3^{2/3}}{3^{-1/3}} &= 3^{2/3 - (-1/3)} \end{aligned} \]

Step 3: Calculate the new exponents

Now, we subtract the fractions to get the new exponents: \[ \begin{aligned} 2^{-1/2 - 3/2} &= 2^{-4/2} \\ 3^{2/3 - (-1/3)} &= 3^{2/3 + 1/3} \end{aligned} \]

Step 4: Simplify the exponents and write the final answer

Lastly, we simplify the exponents further and write the final answer: \[ \begin{aligned} 2^{-4/2} &= 2^{-2}\\ 3^{2/3 + 1/3} &= 3^{3/3} \end{aligned} \] Therefore, the simplified expression using positive exponents only is: \[ \frac{1}{2^2} \cdot 3^1 \Rightarrow \boxed{\frac{3}{4}} \]

Key Concepts

Positive Exponents

Understanding positive exponents is a fundamental concept in algebra and simplifies the handling of mathematical expressions. An exponent indicates how many times a base number is multiplied by itself. For instance, the expression \(5^3\) means \(5 \times 5 \times 5\) which equals 125.

When simplifying expressions, we strive to use positive exponents because they are more straightforward to interpret and work with. A positive exponent, such as \(a^n\) (where 'n' is positive), implies a standard multiplication process. However, when you encounter a negative exponent, the expression can be rewritten with a positive exponent by taking the reciprocal of the base number. This is demonstrated as \(a^{-n} = \frac{1}{a^n}\).

In our textbook exercise, to write the answer using positive exponents only, we must transform negative exponents by using this reciprocal property. For example, \(2^{-2}\) can be written as \(\frac{1}{2^2}\).

Dividing Like Bases

When dividing expressions that have the same base, we can use the exponentiation rule for dividing like bases. This crucial rule states that when you divide powers with the same base, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m - n}\).

Applying this rule makes calculations much more straightforward. For instance, if you have \(\frac{2^3}{2^1}\), we simply subtract the exponents: \(2^{3-1} = 2^2\), which equals 4.

In our example, by dividing like bases of 2 and 3 separately, we significantly simplify the original complex expression. We calculate \(2^{-1/2 - 3/2}\) and \(3^{2/3 - (-1/3)}\), respecting the rule and arriving at new exponents that will lead us to the final simplified expression.

Exponentiation Rules

Exponentiation rules are a set of guidelines that provide a way to handle expressions involving powers efficiently. Some of these rules include the product of powers, quotient of powers as seen in dividing like bases, power of a power, and power of a product.

Here are a few key rules:

• The product of powers rule: \(a^m \cdot a^n = a^{m+n}\)
• The power of a power rule: \((a^m)^n = a^{m\cdot n}\)
• The power of a product rule: \((ab)^n = a^n \cdot b^n\)

These rules serve as shortcuts to simplify expressions without tediously multiplying each base number by itself over and over. In the textbook solution, these rules are applied to combine and simplify the powers of 2 and 3.

Fractional Exponents

Fractional exponents, also known as rational exponents, represent roots and powers using fractions. The expression \(a^{\frac{m}{n}}\) is equivalent to the nth root of \(a^m\), which can also be written as \(\sqrt[n]{a^m}\) or \((\sqrt[n]{a})^m\).

This form of exponentiation is essential when dealing with roots in algebraic expressions. For example, \(8^{\frac{1}{3}}\) represents the cube root of 8, which equals 2. Fractional exponents make it convenient to apply exponentiation rules to roots.

Fractional exponents were utilized in the step-by-step solution of our textbook exercise when combining the exponents of like bases. By calculating expressions like \(2^{-4/2}\) and \(3^{3/3}\), we used the laws governing fractional exponents to simplify the given complex expression to a much more manageable form.