Problem 31 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$\left(m^{2 / 3}\right)\left(m^{5 / 3}\right)$$

Step-by-Step Solution

Answer

The expression simplifies to \( m^{7/3} \).

Step 1: Identify the Operation

Notice that the expression \( (m^{2/3})(m^{5/3}) \) involves multiplication of two terms with the same base \( m \). According to the properties of exponents, we can add the exponents when multiplying like bases.

Step 2: Add the Exponents

Using the rule \( a^m \cdot a^n = a^{m+n} \), we can add the exponents of \( m \):\[ m^{2/3} \cdot m^{5/3} = m^{(2/3 + 5/3)} \]

Step 3: Simplify the Exponents

Add the exponents \( \frac{2}{3} + \frac{5}{3} \):\[ \frac{2}{3} + \frac{5}{3} = \frac{2+5}{3} = \frac{7}{3} \]

Step 4: Write the Final Expression

Now that we have the simplified exponent, represent the expression with positive exponents:\[ m^{7/3} \] Since there are no negative exponents, the expression already satisfies the condition of having only positive exponents.

Key Concepts

Multiplication of Exponents

When you multiply terms that have the same base, you can simplify the expression by applying the properties of exponents. Imagine each term as having a little number "exponent" that tells us how many times to multiply the base by itself. For example, in our exercise, each term is a power of \( m \): \( m^{2/3} \) and \( m^{5/3} \). Since they both have the base \( m \), we can use the multiplication rule for exponents: simply add the exponents together.

• Key Rule: For any base \( a \), multiplying two exponents with the same base means you add the exponents: \( a^m \cdot a^n = a^{m+n} \).
• Example: For \( m^{2/3} \cdot m^{5/3} \), the added exponent is \( \frac{2}{3} + \frac{5}{3} \).

This simplification is what makes working with exponents so powerful and useful in algebraic operations.

Positive Exponents

In algebra, it's often useful and important to express results with positive exponents. A positive exponent means the base is multiplied by itself a certain number of times. In our exercise, all exponents are already positive, which keeps things straightforward and simple. Here are a few reasons why positive exponents are preferred:

• They simplify expressions, avoiding the need for division.
• Positive exponents make it straightforward to evaluate and understand the expression.
• They are directly related to the base power operation, more intuitive in most contexts.

When simplifying expressions, always aim to keep exponents positive to signal that the expression is in its easiest-to-understand form.

Algebraic Operations

Algebraic operations are like the toolkit you use to simplify and manipulate expressions and equations. Knowing how to perform these operations efficiently is key to solving many kinds of math problems. Here is what you need to know about algebraic operations involving exponents:

• Addition/Subtraction: In an expression, exponents can be added when multiplying like bases, and in some cases, exponents can affect subtraction too, but only when dividing similar bases.
• Multiplication: We've seen this with our exponents where exponents add together, making multiplication especially clean when bases match.
• Simplification: Always aim to reduce expressions to their simplest form, which includes rewriting with only positive exponents and combining like terms.

For instance, in our example, multiplying means adding exponents, following the rules, and simplifying neatly. Remember, mastery of these operations lays the foundation for more complex problems in algebra.