Problem 28 - College Algebra

Question

Simplify each exponential expression $$ x^{11} \cdot x^{5} $$

Step-by-Step Solution

Answer

The simplification of the expression \(x^{11} \cdot x^{5}\) is \(x^{16}\).

Step 1: Apply the Product of Powers Property

Given the expression \(x^{11} \cdot x^{5}\), it refers to the multiplication of two powers with the same base (\(x\)). According to the Product of Powers Property, when multiplying two powers with the same base, simply add the exponents. Thus, this expression simplifies to \(x^{11+5}\).

Step 2: Add the Exponents to Simplify the Expression

Add the exponents 11 and 5 to simplify the expression, resulting in \(x^{16}\).

Key Concepts

Product of Powers Property

The Product of Powers Property is a fundamental rule in mathematics used to simplify expressions involving exponents. When multiplying two expressions with the same base, this property allows us to keep that base and simply add the exponents. This is because the base number is being multiplied by itself multiple times across both terms. For example, in the expression \(x^{11} \cdot x^{5}\), both terms have the base \(x\). According to the Product of Powers Property, you add the exponents: \(x^{11+5}\). Remember, this only applies when the bases are identical, so make sure both bases are the same before applying this rule.

This property helps us simplify expressions easily, especially when dealing with large numbers or more complex algebraic expressions. Always check that you're applying this property correctly by verifying that the bases are the same.

multiplication of powers

When we multiply powers, we're actually performing repeated multiplication. Specifically, we're taking one power and multiplying it by another power. In mathematics, powers are also referred to as exponents. The multiplication of powers becomes straightforward with rules like the Product of Powers Property, which provides a quick way to handle the multiplication when bases are the same.

Consider \(x^{11} \cdot x^{5}\):

• Both parts of the expression (\(x^{11}\) and \(x^{5}\)) have the same base, \(x\).
• Using the Product of Powers Property, the multiplication of these powers is simplified to adding the exponents.

This allows you to transform a complex multiplication of powers into a simpler form, such as \(x^{16}\). This simplification makes calculations easier and more manageable, especially in algebra and higher mathematics.

adding exponents

Adding exponents occurs in multiplication scenarios where powers share a common base. Understanding this is crucial for simplifying exponential expressions efficiently. Whenever you see two powers with the same base being multiplied, you add their exponents. This is not true for all operations, however, only multiplication.

Take a closer look at \(x^{11} \cdot x^{5}\):

• Notice that the base, \(x\), remains constant.
• Rather than computing \(x^{11}\) and \(x^{5}\) separately, simply add the exponents: \(11 + 5 = 16\).

So, the expression simplifies to \(x^{16}\). Remember to only add exponents directly when dealing with multiplication of like bases. Practice using this technique by identifying the base and recognizing when exponents can be combined smoothly, reinforcing this key algebraic concept.