Problem 23 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ 5^{8} \cdot 5^{3} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(5^{11}\)

Step 1: Identify the base and the exponents

From the given expression \(5^{8} \cdot 5^{3}\), we see that the base is 5 and we have two exponents, 8 and 3.

Step 2: Apply exponent laws

Using the exponent law that states when multiplying powers with the same base, add the exponents, the expression becomes \(5^{8+3}\).

Step 3: Simplify the expression

Simplify the expression \(5^{8+3}\) to find the final solution. Adding the exponents yields \(5^{11}\).

Key Concepts

Multiplying Powers

Multiplying powers refers to a fundamental rule when dealing with exponents. When you multiply powers, it means you have a situation where two or more exponential expressions share the same base. To solve it efficiently, you can apply one of the exponential laws, which is designed to simplify the operation.
When you multiply powers that have the same base, the rule is simple: you add the exponents while keeping the base unchanged.

• If you have two numbers in the form of \(a^m\) and \(a^n\), where both have 'a' as the base, the product can be expressed as \(a^{m+n}\).

In this case, suppose you have the expression \(5^8 \cdot 5^3\).
By applying the multiplication rule, you simply add the exponents 8 and 3, resulting in \(5^{8+3}\).

Base and Exponents

Understanding the components of powers is crucial in mastering exponents. A power consists mainly of two parts: the base and the exponent.
The base is the number that will be multiplied by itself a certain number of times, which is indicated by the exponent.

• For example, in the expression \(5^8\), '5' is the base.
• The exponent, which is '8' in this case, tells you how many times the base is used as a factor in the multiplication.

In the given exercise \(5^8 \cdot 5^3\), '5' serves as the base for both terms. Exponents '8' and '3' tell us that '5' is multiplied by itself 8 times, and 3 times, respectively.
So, the operation essentially means multiplying the same base a total of \(8 + 3\) times when the powers are combined.

Simplifying Expressions

Simplifying expressions involving exponents is the art of making a mathematical expression shorter or easier to understand.
In the exercise, once you identify the base and know the operation for multiplying powers, the next step is simplification. You aim to combine and reduce when possible to find a more straightforward form.

• In the expression \(5^{8+3}\), you simplify by actually performing the addition in the exponent: \(8 + 3 = 11\).
• This reduces the expression to a single power: \(5^{11}\).

By simplifying, you go from two multiplication terms to one concise power term, making calculations easier and the expression more accessible.