Problem 64 - Beginning and Intermediate Algebra

Question

Convert to scientific notation. $$5.87 \times 10^{8}$$

Step-by-Step Solution

Answer

The given number \(5.87 \times 10^8\) is already in scientific notation with \(a = 5.87\) and \(b = 8\).

Step 1: Identify the Conversion

We need to convert: \(5.87 \times 10^{8}\).

Step 2: Apply the Conversion Rule

We apply the appropriate conversion formula or rule.

Step 3: State the Result

The given number \(5.87 \times 10^8\) is already in scientific notation with \(a = 5.87\) and \(b = 8\).

Key Concepts

Understanding Exponents

Exponents are a way to represent repeated multiplication of the same number. They simplify long expressions, making it easier to read and understand them. For example, instead of writing 10 multiplied by itself eight times, we can write it as \(10^8\). This means we are using 10 as a factor eight times. The base, 10 in this example, is the number being multiplied. The exponent, which is 8, tells us how many times the base is used as a factor.

• Base: The number that is being multiplied.
• Exponent: The number of times the base is multiplied by itself.

Exponents are pivotal in scientific notation as they allow for the compact representation of very large or very small numbers.

What Is an Integer?

An integer is a number that has no fractional or decimal part. Integers can be positive, negative, or zero. They are whole numbers such as -3, 0, and 7. In the context of scientific notation, the exponent is always an integer because it needs to express whole power levels of 10.
Using integers, especially for exponents in scientific notation, helps maintain the precision of the number representation. For instance, in \(5.87 \times 10^8\), 8 is the integer exponent that conveys how many times 10 should be multiplied to get our desired scale.

What is Standard Form in Scientific Notation?

Standard form is a way of writing numbers that are too large or too small to be conveniently written in decimal form. In scientific notation, standard form is expressed as \(a \times 10^b\), where \(1 \leq a < 10\) and \(b\) is an integer. This form helps simplify calculations and makes reading very large or small values intuitive.
The number \(5.87 \times 10^8\) is already in standard form. Here, \(a = 5.87\) lies within the required range (between 1 and 10), and \(b = 8\) is an integer, fulfilling the conditions for proper scientific notation.

How Multiplication Works with Scientific Notation

Multiplication in scientific notation involves multiplying the base numbers and adding their exponents. This makes computation more efficient, especially with very large or small numbers.For example, when multiplying \( (2 \times 10^3) \times (3 \times 10^4) \):

• Multiply the coefficients: \(2 \times 3 = 6\)
• Add the exponents: \(3 + 4 = 7\)

This results in \(6 \times 10^7\). This approach leverages the laws of exponents which state \(a^m \times a^n = a^{m+n}\). The same principles apply no matter how large or small the numbers are, ensuring consistent and quick calculations.