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Chapter 10: Problem 17

Multiply. $$5 \cdot 5^{2} \cdot 5^{9}$$

Short Answer

Expert verified
The result is \(5^{12}\).

Step by step solution

01

Identify the Base

All the terms have the same base, which is 5.
02

Apply the Product of Powers Property

Use the product of powers property, which states that when multiplying powers with the same base, you add the exponents. So, we have \(5^1 \times 5^2 \times 5^9\).
03

Add the Exponents

Add the exponents: \(1 + 2 + 9 = 12\).
04

Write the Final Expression

Combine the base and the new exponent: \(5^{12}\).

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Key Concepts

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

product of powers property
When working with exponents, the product of powers property is super useful. This property says that if you are multiplying numbers with the same base, you simply add the exponents together. For example, if you have something like \(a^m \times a^n\), you can combine those into \(a^{m+n}\). This makes multiplication of powers much simpler and helps in quickly reaching the final solution. Imagine you have \(5^1 \times 5^2 \times 5^9\); with the product of powers property, this can be simplified to \(5^{1+2+9}\), which is \(5^{12}\).
multiplication
Multiplication in algebra can sometimes be intimidating, especially with exponents. However, by understanding the rules and properties, it becomes much more manageable. When we multiply numbers, we're just adding groups of a number. For example, \(2 \times 3\) means adding 2 three times (\( 2 + 2 + 2 \)). Similarly, when multiplying exponents with the same base, you follow the product of powers property to make the process simpler. So \(5 \times 5^2 \times 5^9\) becomes a straightforward application of the product of powers property.
base and exponent
In an expression involving exponents like \(5^2\), 5 is called the base, and 2 is the exponent. The base is the number being multiplied by itself, while the exponent tells you how many times to multiply the base by itself. For instance, \(5^2\) means \(5 \times 5\), and \(5^3\) would mean \(5 \times 5 \times 5\). Understanding the base and exponent helps in identifying patterns and simplifying expressions, especially when applying the product of powers property.
simplifying expressions
Simplifying expressions makes complex equations easier to work with and understand. To simplify an expression with exponents, start by identifying any common bases. Use properties like the product of powers property to combine these terms. For instance, take \(5 \times 5^2 \times 5^9\). Here, the base is 5. By applying the product of powers property, add the exponents (remember that \(5\) is \(5^1\)) to get \(5^{1+2+9} = 5^{12}\). Simplifying helps in finding solutions faster and makes calculations more manageable.

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