Problem 103 In the following exercises, simp... [FREE SOLUTION]

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Chapter 1: Problem 103

In the following exercises, simplify each expression. $$ 5^{3} $$

Short Answer

Expert verified

125

Step by step solution

Understand the Expression

The expression given is a power expression, where the base is 5 and the exponent is 3.

Apply the Power Rule

According to the power rule, when you have a base raised to an exponent, you multiply the base by itself as many times as the value of the exponent. This means we need to calculate 5 multiplied by itself 3 times.

Perform the Multiplication

Multiply 5 by itself three times: \[ 5^3 = 5 \times 5 \times 5 \]Calculate each step: \[ 5 \times 5 = 25 \]Now multiply the result by 5: \[ 25 \times 5 = 125 \]

Simplified Expression

The simplified expression of $5^3$ is 125.

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Key Concepts

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

Power Rule

The power rule is a fundamental concept when working with exponents. It states that when a number (called the base) is raised to a power (called the exponent), you multiply the base by itself as many times as the value of the exponent. For example, in the expression $5^3$, 5 is the base and 3 is the exponent. When you apply the power rule, you multiply 5 by itself three times: \[ 5 \times 5 \times 5 \]. This helps you understand what the exponent indicates. The rule makes complex multiplication easier and more structured.

Base and Exponent

The base and exponent are crucial parts of an expression involving exponents. The base is the number we are multiplying. The exponent tells us how many times to multiply that base. In the expression $5^3$, the number 5 is the base, and the number 3 is the exponent. Understanding the roles of both is important:

Base: This is the number you keep multiplying (5 in this case).
Exponent: This tells you how many times to multiply the base by itself (3 times for this example).

You can think of the exponent as a way to simplify repeated multiplication. Instead of writing \[ 5 \times 5 \times 5 \], you simply write $5^3$.

Multiplication of Powers

Multiplication of powers follows specific rules, especially when the bases are the same. Generally, you combine the exponents when multiplying like bases. For example, if you have two expressions like $a^m \times a^n$, you add the exponents: $a^{m+n}$. However, if you're just working with a single power, like $5^3$, you directly multiply the base by itself as indicated by the exponent. In this case, it results in:

5 multiplied by 5 equals 25.
Then, 25 multiplied by 5 equals 125.

Understanding these steps helps break down more complicated expressions where multiple powers and bases are involved.

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91影视

In the following exercises, simplify each expression. $$ 5^{3} $$

Short Answer

Step by step solution

Understand the Expression

Apply the Power Rule

Perform the Multiplication

Simplified Expression

Key Concepts

Power Rule

Base and Exponent

Multiplication of Powers

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