Problem 46 - Algebra A Combined Function

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ 5^{1 / 2} \cdot 5^{1 / 6} $$

Step-by-Step Solution

Answer

The expression simplifies to \(5^{2/3}\).

Step 1: Identify the Base

Notice that both terms in the expression \(5^{1/2} \cdot 5^{1/6}\) have the same base of \(5\). This allows us to use the properties of exponents to combine the terms.

Step 2: Apply the Product of Powers Property

The product of powers property states that when you multiply two powers that have the same base, you add the exponents: \(a^m \cdot a^n = a^{m+n}\). For our expression, this means:\[5^{1/2} \cdot 5^{1/6} = 5^{(1/2 + 1/6)}.\]

Step 3: Add the Exponents

To add \(1/2\) and \(1/6\), first find a common denominator. The least common denominator of 2 and 6 is 6. Convert \(1/2\) to \(3/6\), and \(1/6\) remains \(1/6\). Add the fractions:\[\frac{3}{6} + \frac{1}{6} = \frac{4}{6}.\] Simplify \(\frac{4}{6}\) to \(\frac{2}{3}\).

Step 4: Write the Final Answer with Positive Exponents

Substitute the simplified exponent back into the expression from Step 2 to get the answer:\[5^{4/6} = 5^{2/3}.\] This is already expressed with a positive exponent, as required.

Key Concepts

Product of Powers Property

The product of powers property is a fundamental concept in mastering exponents. This property states that when you multiply two exponents with the same base, the exponents can be added together:

• For example, if you have an expression like \( a^m \cdot a^n \), the result is \( a^{m+n} \).

This allows for a more straightforward way to simplify expressions that involve identical bases. It works the same whether the exponents are whole numbers, fractions, or any other real number.
The key is that the base of the terms being multiplied must be the same, making it easy to combine them by simply adding their exponents.
For instance, given the expression \( 5^{1/2} \cdot 5^{1/6} \), you first recognize that the bases are both 5. Then, you apply the product of powers property and add the exponents \( \frac{1}{2} + \frac{1}{6} \) to get the new exponent value. Remember to first convert any fractions to have a common denominator before performing the addition.

Simplifying Expressions

Simplifying expressions with exponents can make your life a lot easier and help you solve mathematical problems more efficiently. When you come across expressions like \( 5^{1/2} \cdot 5^{1/6} \), simplifying them by using exponent rules can lead you to the neatest answer.
Following the steps properly is crucial:

• First, identify the common base of the exponents, which makes it possible to apply the product of powers property.
• Next, when adding fractions like \( \frac{1}{2} \) and \( \frac{1}{6} \), a common denominator is necessary. In this case, it would be 6.
• Convert \( \frac{1}{2} \) to \( \frac{3}{6} \) and proceed to add the two fractions \( \frac{3}{6} + \frac{1}{6} \).

This results in an exponent of \( \frac{4}{6} \), which can be simplified further to \( \frac{2}{3} \).
This simplicity makes it easier to understand and manage complex mathematical expressions by reducing them to the most direct form. Make sure always to express your final answer with positive exponents.

Positive Exponents

Expressing exponents positively is often preferred in mathematics because it simplifies understanding and interpretation of results.
Positive exponents imply repeated multiplication of the base. In our example of \( 5^{2/3} \), this means multiplying \( 5 \) by itself two-thirds of a time.
This is much more intuitive than dealing with negative exponents, which suggest division or reciprocals.
To ensure that expressions are expressed with positive exponents, always follow through with reducing fractions to their simplest form, like simplifying \( \frac{4}{6} \) to \( \frac{2}{3} \).

• This ensures clarity and keeps your expression as straightforward as possible for anyone looking at your work.
• Remember that in cases where you start with a negative exponent, you can convert it by taking the reciprocal of the base. However, in exercises requiring positive exponents only, focus on managing and simplifying the original terms carefully to stay within the positive range, especially with fractional exponents.

Keeping exponents positive is a hallmark of a well-simplified expression.