Problem 1 Complete the rule for multiplyin... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

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Chapter 10: Problem 1

Complete the rule for multiplying like bases. $a^{m} a^{n}=$ ____.

Short Answer

Expert verified

The rule is $ a^{m} a^{n} = a^{m+n} $.

Step by step solution

Understand the Rule

Recognize that you are multiplying two expressions that have the same base, which is 'a' in this case. The bases are raised to different exponents 'm' and 'n'.

Recall the Property of Exponents

Recall the exponent rule for multiplying like bases: \ If you have two expressions with the same base, you can add their exponents when you multiply them. Mathematically, this property is written as: \ \[ a^{m} \times a^{n} = a^{m+n} \]

Apply the Rule

Apply the rule to the given expression $ a^{m} \times a^{n} $ by adding the exponents: \ \[ a^{m} \times a^{n} = a^{m+n} \]

Write the Final Answer

After applying the exponent rule, the expression becomes $ a^{m+n} $.

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Key Concepts

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

Like Bases

When working with exponents, 'like bases' refer to expressions that have the same base number or variable. For instance, in the expressions $a^m$ and $a^n$, the base is 'a'. Because the base is the same in both expressions, they are considered 'like bases'. Understanding that the bases are identical is crucial when applying exponent rules, especially when multiplying or dividing exponential expressions.

Exponent Rules

Exponent rules are guidelines that simplify operations involving exponents. One of the most important rules for multiplying exponents with like bases is: if you multiply two expressions with the same base, you add their exponents. Mathematically, this rule is expressed as:
\[ a^m \times a^n = a^{m+n} \]
This rule works because exponents represent repeated multiplication. For example, $a^3 \times a^2$ means $a \times a \times a \times a \times a$, which can be rewritten as $a^{3+2} = a^5$.

Addition of Exponents

When multiplying like bases, you need to add the exponents. This process is essential for simplifying expressions correctly. Let's break this down step by step:

Consider the expression $a^m \times a^n$. The base 'a' remains the same.
The exponents 'm' and 'n' are added together, resulting in $a^{m+n}$.
This addition is based on the rule that exponentiation is repeated multiplication, so combining two sets of exponents means adding them up.

Always remember to add the exponents only when the bases are the same. This rule helps in making sense of complex exponentials and simplifies equations effectively.

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91影视

Complete the rule for multiplying like bases. \(a^{m} a^{n}=\) ____.

Short Answer

Step by step solution

Understand the Rule

Recall the Property of Exponents

Apply the Rule

Write the Final Answer

Key Concepts

Like Bases

Exponent Rules

Addition of Exponents

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