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Chapter 5: Problem 74

Simplify the expression. Write your answer using only positive exponents. $$ 2^3 \cdot 2^2 $$

Short Answer

Expert verified
The simplified expression is \(2^5\).

Step by step solution

01

Identify the base and exponents

Start by identifying that the base of both expressions is 2. The first exponent is 3 and the second is 2.
02

Apply the law of exponents

Use the law of exponents which states that when you multiply bases of the same value, you can add the exponents. So, \(2^3 \cdot 2^2 = 2^{3+2}\).
03

Simplify the expression

Next, simplify the expression by adding the exponents together. \(2^{3+2} = 2^5\).

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Key Concepts

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

The Law of Exponents
The law of exponents provides a simple method for handling expressions that involve powers or exponents. It is a crucial rule in algebra that helps simplify complex expressions. One key rule is that when multiplying two expressions with the same base, you can simply add the exponents.
For example, in the expression \(2^3 \cdot 2^2\), both terms have the base of 2. The law of exponents tells us to add the exponents: thus, \(2^3 \cdot 2^2 = 2^{3+2}\).
This results in \(2^5\).
  • Multiplying with the Same Base: Add the exponents.
  • Example: \(a^m \cdot a^n = a^{m+n}\).

The law of exponents makes it easier to work with powers by leveraging simple addition instead of complicated multiplication.
Simplifying Expressions
Simplifying expressions is an essential algebraic skill that makes problems easier to solve. It involves reducing complex expressions into a more manageable form. When working with exponents, simplifying often means applying laws of exponents to consolidate terms.
In our example, \(2^3 \cdot 2^2\), we simplified the expression using the law of exponents. By adding the exponents, the expression reduced from its original form to \(2^5\).
Let's break that down:
  • Start with Identifying: Look for common bases.
  • Apply Appropriate Laws: Use rules such as adding exponents for multiplication.
  • Simplify Further if Possible: Once you have the expression with positive exponents, calculate if necessary.

Simplifying involves these straightforward steps, turning a complex expression into an easier-to-handle form.
Positive Exponents
When dealing with exponents, it's often desirable to express results with positive exponents. A positive exponent signifies that the base is multiplied by itself a certain number of times, which is straightforward to interpret.
For example, \(2^5\) means that the number 2 is multiplied by itself five times.
  • Positive Exponent Meaning: Indicates repeated multiplication.
  • Converting Negative to Positive: Reciprocal of the base can turn a negative exponent positive.

Positive exponents provide a clear indication of the size or magnitude of a number, making expressions simpler and more intuitive to work with. When the exercise requires you to "write your answer using only positive exponents," it's ensuring the form is ready for clear interpretation.

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