Problem 45 - Algebra A Combined Function

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ 3^{1 / 4} \cdot 3^{3 / 8} $$

Step-by-Step Solution

Answer

The expression simplifies to \(3^{5/8}\).

Step 1: Identify like bases

The expression given is \( 3^{1/4} \cdot 3^{3/8} \). Both terms have the same base, which is 3.

Step 2: Apply the Product of Powers Property

When multiplying expressions with the same base, you add the exponents. This is known as the product of powers property. Here we have: \[3^{1/4} \cdot 3^{3/8} = 3^{(1/4) + (3/8)}\]

Step 3: Find a common denominator for the exponents

Since the exponents \(1/4\) and \(3/8\) have different denominators, we need to find a common denominator to add them. The least common denominator of 4 and 8 is 8. Convert \(1/4\) to \(2/8\): \[1/4 = 2/8\]

Step 4: Add the exponents

Now add the exponents with the same denominator: \[3^{(2/8) + (3/8)} = 3^{5/8}\]

Step 5: Write with positive exponents

Since the question already asks for positive exponents, the final expression \(3^{5/8}\) is already expressed with a positive exponent.

Key Concepts

Product of Powers Property

When you multiply expressions that have the same base, there's a handy property to use called the Product of Powers Property. This property tells us that we can simply add the exponents together. For instance, if you have an expression like \(3^a \cdot 3^b\), you can add the exponents \(a\) and \(b\) to get \(3^{a+b}\).

• This property only works if the bases are the same. For example, you cannot use it on \(2^3 \cdot 3^4\) as the bases differ.
• The result is faster and more efficient, especially with larger expressions.

To apply this in a math problem, like \(3^{1/4} \cdot 3^{3/8}\), note both terms have the same base of 3, allowing you to add the exponents \(1/4\) and \(3/8\). This simplifies it to \(3^{(1/4) + (3/8)}\). Be sure not to miss the same base condition, or it won't work!

Finding a Common Denominator

Adding fractions like exponents sometimes means dealing with different denominators. That's where finding a common denominator comes in. To add fractions directly, they need to have the same denominator.
For example, in \(1/4\) and \(3/8\), the denominators are different: 4 and 8. The least common denominator here is 8, which means converting both fractions to have this denominator.

• Convert \(1/4\) to \(2/8\) as follows: Multiply both the numerator and denominator by 2.

This conversion lets you add \(2/8\) and \(3/8\) directly, simplifying to \(5/8\).
Finding common denominators is a crucial skill in fractions, beyond just exponents. It lets you handle calculations involving addition, making your math straightforward and reliable.

Simplifying Expressions

Simplifying expressions is about making them as clear and simple as possible. In the context of exponents, it means reducing the expression to its most straightforward form, using positive exponents. Often, this involves using various mathematical rules and properties.
Here's a breakdown of the simplification steps:

• Start by applying properties like the Product of Powers, as covered earlier.
• Find a common denominator if the exponents include fractions, allowing you to add them easily.
• Finally, ensure your result is expressed with positive exponents if required.

With our expression \(3^{5/8}\), the final step was simply ensuring the exponent is positive, which it already was.
Overall, knowing how to simplify expressions keeps things neat and reduces errors in calculations, letting you focus on essential numerical operations.