Problem 29 - College Algebra

Question

Simplify each exponential expression $$ x^{-5} \cdot x^{10} $$

Step-by-Step Solution

Answer

The simplified form of the expression \( x^{-5} \cdot x^{10} \) is \( x^{5} \).

Step 1: Identify the base and exponents

We identify 'x' as our base and -5 and 10 as our exponents. The expression is \( x^{-5} \cdot x^{10} \).

Step 2: Use the rules of exponents

When we multiply expressions with the same base, we add the exponents. This rule is given as \( a^{n} \cdot a^{m} = a^{n+m} \). Applying this, we get \( x^{-5 + 10} = x^{5} \).

Step 3: Simplify the expression

The final simplified expression for \( x^{-5} \cdot x^{10} \) is \( x^{5} \).

Key Concepts

Rules of Exponents

Exponents can seem tricky at first, but they follow some simple, consistent rules that make working with them easier. One of the key rules is that when you multiply exponential expressions with the same base, you simply add their exponents together.
For example, if you have an expression such as \( a^m \cdot a^n \), you can simplify this to \( a^{m+n} \). This is because multiplying powers with the same base is equivalent to adding the number of times you multiply the base by itself.

• Multiplication Rule: For any non-zero number \( a \), and any integers \( m \) and \( n \), \( a^m \cdot a^n = a^{m+n} \).

You can also use other exponent rules like the division rule, which states that when dividing, you subtract the exponents: \( a^m / a^n = a^{m-n} \). Knowing these simple rules allows you to simplify many complex expressions very quickly.

Simplifying Expressions

When simplifying expressions, the goal is to make them as easy to understand and as simple as possible. Simplifying often involves using the rules of exponents to combine like terms when possible.
For the original exercise, you had the expression \( x^{-5} \cdot x^{10} \). By using the rule of exponents for multiplication, you combined the exponents because the base \( x \) was the same in both terms.

• Combine Like Terms: When you see terms with the same base and different exponents being multiplied, add the exponents together.
• Result: From the original expression, using \( x^{-5} \cdot x^{10} = x^{-5+10} = x^{5} \), you simplify the expression to \( x^5 \).

This method reduces the complexity of expressions, making it easier to handle more complex equations later.

Multiplication of Exponents

The multiplication of exponents follows straightforward rules. When dealing with exponents, the base number is raised to a certain power by indicating how many times you multiply the base number by itself.
In the context of multiplying exponents, when two expressions with the same base are multiplied together, you just add their exponents.
Let's break this down:

• Same Base: Make sure both exponential expressions have the same base.
• Add Exponents: Simply add the powers together, \( a^m \cdot a^n = a^{m+n} \).

Thus, in the expression \( x^{-5} \cdot x^{10} \), you add the powers \(-5\) and \(10\) to get \(5\). As a result, combining these expressions gives you \( x^5 \), simplifying your calculation substantially.