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Chapter 5: Problem 19

\(\left(4 \times 10^{3}\right)\left(2 \times 10^{4}\right)\)

Short Answer

Expert verified
8 \times 10^7

Step by step solution

01

- Identify Base Numbers and Exponents

Identify the base numbers and their exponents in each term. In this problem, the first term is written as \(4 \times 10^3\) and the second term is \(2 \times 10^4\). Here, the base numbers are 4 and 2, and the exponents of 10 are 3 and 4 respectively.
02

- Multiply the Base Numbers

Multiply the base numbers (4 and 2) together. This gives \(4 \times 2 = 8\).
03

- Add the Exponents

Since the base of the exponents is the same (10), add the exponents together: \(3 + 4 = 7\).
04

- Combine the Results

Combine the multiplied base result and the added exponents to form the final answer: \(8 \times 10^7\).

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Key Concepts

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

base numbers
In scientific notation, the base number is the number you multiply by a power of 10.
For example, in the term \(4 \times 10^3\), the base number is 4.
Base numbers can be any real number, but in scientific notation, they are typically between 1 and 10.
Identifying the base number helps you separate it from the exponent part of the notation.
For instance, in the exercise \(\left( 4 \times 10^3 \right) \left( 2 \times 10^4 \right)\), the base numbers are 4 and 2.
exponents
Exponents are used to show how many times a number, called the base, is multiplied by itself.
In our exercise, the exponent 3 in \(4 \times 10^3\) means \(10\) is multiplied by itself three times: \(10 \times 10 \times 10\).
When multiplying numbers in scientific notation, if the base of the exponents is the same, you add the exponents together.
So in \(\left ( 4 \times 10^3 \right) \left ( 2 \times 10^4 \right ) \), the exponents 3 and 4 are added to give 7.
scientific notation
Scientific notation is a way to express very large or very small numbers.
It uses base numbers and exponents, simplifying calculations and easily representing numbers.
Standard form is \(a \times 10^n\) where a is the base number and n is the exponent.
For example, \(4 \times 10^3\) means 4 times 10 to the power of 3, which equals 4000.
Likewise, \(2 \times 10^4\) equals 20000.
This notation makes it easier to handle very large or small values in calculations.
multiplication
To multiply numbers in scientific notation, start with the base numbers and then the exponents.
First, multiply the base numbers independently.
In our exercise, \(4 \times 2 = 8\).
Next, add the exponents because the base of the exponent is the same (10).
So, \(3 + 4 = 7\).
Finally, combine the results to get \(8 \times 10^7\).
This method simplifies multiplying large numbers into manageable steps.

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