Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Carry out the indicated operation and write your answer using positive exponents only. $$ \frac{x^{7 / 3}}{x^{-2}} $$

Step-by-Step Solution

Answer

The simplified expression with positive exponents only is \(x^{13/3}\).

Step 1: Identify Exponent Properties

Observe that we are given a fraction, and we need to simplify the expression using exponent properties. We will mainly use the property \(a^{m} \div a^{n} = a^{(m-n)}\).

Step 2: Apply Exponent Properties

Using the property mentioned in step 1, apply it to the given expression: \(\frac{x^{7 / 3}}{x^{-2}} = x^{(7/3)-(-2)}\)

Step 3: Simplify the Exponent

Now, we will simplify the exponent by combining the fractions: \((7/3) - (-2) = (7/3) + (6/3) = (7+6)/3 = 13/3\) So, the simplified expression becomes: \(x^{(7/3)-(-2)}= x^{13/3}\)

Step 4: Final Answer

By simplifying the expression using exponent properties and simplifying the exponent term, the final answer is: \(x^{13/3}\)

Key Concepts

Fraction Simplification

Fraction simplification in mathematics involves reducing a fraction to its simplest form. This form is often achieved by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number. However, when working with variables and expressions involving exponents, simplification takes on a different approach.
When dealing with algebraic fractions that include exponents, you often use the properties of exponents to simplify. This means handling variable bases with exponent rules, like subtracting the powers when dividing similar bases. Because these expressions operate under the same principles as numerical fractions, understanding basic fraction operations helps greatly in tackling more complex equations.
Remember that the key goal in simplification is to make the expression as straightforward as possible while keeping all the mathematical relationships intact. This not only helps in presenting clearer solutions but also eases the process when further calculations or substitutions are necessary.

Exponent Properties

Exponent properties are essential tools in algebra that allow us to manipulate and simplify expressions involving powers. A crucial exponent rule is the quotient of powers property, which states that when dividing two exponents with the same base, you subtract the exponent in the denominator from the exponent in the numerator:

• \( a^m \div a^n = a^{(m-n)} \)

These properties help in converting expressions to a simpler form, making them easier to work with, especially when more operations are required later.
In the example provided,

• We are given \( \frac{x^{7/3}}{x^{-2}} \), where the exponent in the numerator is \(7/3\) and the exponent in the denominator is \(-2\).
• Applying the property, we rewrite the expression as \( x^{(7/3) - (-2)} \).

This shortens the expression considerably, avoiding cumbersome calculations and allowing us to spot patterns or solutions swiftly. Always remember, when you encounter negative exponents, these just indicate reciprocals and can often be handled smoothly by conversion to positive exponents.

Mathematical Operations

Mathematical operations in algebra often require a mix of performing arithmetic calculations and applying algebraic properties. Here, combining arithmetic operations with exponent rules is the key.
Consider the operation \((7/3) - (-2)\). Initially, the task is to convert the subtraction into an addition because subtracting a negative is the same as adding the positive equivalent:

• \((7/3) + 2\)

Next, it's crucial to express the whole number \(2\) as a fraction to easily carry out the addition:

• \(2 = \frac{6}{3}\)
• So, \((7/3) + (6/3) = (13/3)\).

Once computed, the final resultant exponent becomes \(13/3\), which is easier to interpret and can be written using all positive exponents:

• \(x^{13/3}\)

By mastering small arithmetic steps combined with understanding exponent rules, mathematical operations involving exponents become much more manageable and less prone to errors.