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Chapter 10: Problem 345

In the following exercises, simplify. $$r^{-2} \cdot r^{5}$$

Short Answer

Expert verified
The simplified form is \( r^{3} \).

Step by step solution

01

Understand the properties of exponents

When multiplying expressions that have the same base, the exponents should be added. This is based on the property of exponents: \[ a^m \times a^n = a^{m+n} \].
02

Apply the property to the given expression

In the expression \( r^{-2} \times r^5 \), identify the exponents of the same base \(r\), which are \(-2\) and \(5\). Add these exponents together: \[ -2 + 5 = 3 \].
03

Write the simplified expression

Combine the result from the previous step with the base \(r\) to get: \[ r^{3} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Exponents
When working with exponents, understanding their properties is crucial for simplifying expressions. One fundamental property is related to multiplying expressions with the same base. The rule is as follows:
  • When you multiply two exponential expressions with the same base, you add their exponents: \[ a^m \times a^n = a^{m+n} \]
  • For example, if you have \[ x^2 \times x^3 \], you add the exponents (2 and 3) to get \[ x^5 \].
Remember, this property only works when the base is the same for both expressions. This concept helps us understand and simplify more complex expressions.
Multiplying Exponents
Multiplying exponents is straightforward if you follow the rules. Let's break it down:
  • Identify the base of the expressions. Make sure they are the same.
  • Look at the exponents that correspond to each base.
  • Add the exponents together.
For example, in the expression \[ r^{-2} \times r^5 \], both terms have the same base, which is \[ r \]. The exponents are \[-2 \] and \[ 5 \]. Using the property from before, add these exponents:
\[ -2 + 5 = 3 \]
Thus, the product is \[ r^3 \]. By using the properties of exponents systematically, you can simplify even the trickiest expressions.
Simplifying Expressions
Simplifying expressions with exponents often involves combining several steps. Here's a general approach:
  • Identify if you can use any properties of exponents such as multiplication, division, or power of a power.
  • Apply the appropriate property to combine the exponents.
  • Write the simplified expression.
In our example, we had \[ r^{-2} \times r^5 \]. By identifying the property of adding exponents when multiplying, we simplified the expression to \[ r^3 \].
This is a straightforward process but can become easier with more practice. Always remember to look out for like bases and apply the relevant properties of exponents. With these tools, you can simplify complex exponent expressions with ease.

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