Problem 37 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Carry out the indicated operation and write your answer using positive exponents only. $$ x^{2 / 5}\left(x^{2}-2 x^{3}\right) $$

Step-by-Step Solution

Answer

The short answer is \(x^{\frac{12}{5}} - 2x^{\frac{17}{5}}\).

Step 1: Distribute

Since we are given an expression in the form \(x^{2/5} (x^2 - 2x^3)\), our first step is to distribute \(x^{2/5}\) to both terms inside the parentheses. \[ x^{2 / 5}\left(x^{2}\right) - x^{2 / 5}\left(2 x^{3}\right) \]

Step 2: Apply Exponent Rules

We can simplify this expression using the exponent rules. To multiply two powers with the same base, we add the exponents. \[ x^{(2 / 5) + 2} - 2x^{(2 / 5) + 3} \]

Step 3: Simplify the Exponents

Now, we simplify the exponents by adding the fractions and integers. \[ x^{\frac{12}{5}} - 2x^{\frac{17}{5}} \]

Step 4: Write Answer with Positive Exponents

As the final answer already has positive exponents, we do not need to make any further adjustments: \[ x^{\frac{12}{5}} - 2x^{\frac{17}{5}} \]

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of symbols that represent numbers and operations. In algebra, we often work with variables like \( x \), which can take various values. An algebraic expression consists of terms that are added or subtracted. Each term is usually a product of a number, known as a coefficient, and a variable raised to a power, which is the exponent.

For instance, in the expression \( x^{2/5} (x^2 - 2x^3) \), the term \( x^{2/5} \) is multiplied by the binomial \( (x^2 - 2x^3) \). This expression involves two key mathematical concepts: multiplication of terms and the use of exponents.

In such expressions, it is crucial to pay attention to:

• The coefficients, which tell us how many of each term we have.
• The exponents, which dictate how many times the base (the variable) is multiplied by itself.

Understanding how to manipulate these parts of an expression leads to effectively solving algebraic problems.

Distributive Property

The distributive property is a powerful tool in algebra that allows us to simplify expressions. It states that multiplying a number by a sum or difference is the same as multiplying each term separately and then adding or subtracting the results. In formula form, it can be seen as \( a(b + c) = ab + ac \).

In our exercise, we apply this property to \( x^{2/5}(x^2 - 2x^3) \). Here, \( x^{2/5} \) is distributed to each term inside the parentheses:

• First, multiply \( x^{2/5} \) by \( x^2 \) to get \( x^{(2/5) + 2} \).
• Next, multiply \( x^{2/5} \) by \( 2x^3 \) to get \( 2x^{(2/5) + 3} \).

Simplifying each product gives us: \( x^{12/5} \) and \( -2x^{17/5} \). By distributing and simplifying in this way, we transform the original expression into a more easily computed form.

Simplifying Fractions

When working with exponents, especially with fractional exponents, simplifying fractions is a crucial step. Fractional exponents represent powers and roots. For instance, \( x^{2/5} \) means we take the fifth root of \( x \) and then square it.

In our solution, we added fractions while applying the exponent rules. When adding \( x^{(2/5) + 2} \) and \( x^{(2/5) + 3} \), we turned the integers into fractions: 2 becomes \( \frac{10}{5} \) and 3 becomes \( \frac{15}{5} \).

Combining these gives us:

• For the first term: \( x^{(2/5) + 2} = x^{12/5} \).
• For the second term: \( x^{(2/5) + 3} = x^{17/5} \).

Understanding how to add and simplify fractions allows us to easily handle expressions that involve fractional exponents, thereby simplifying the overall calculation.