Problem 67 - Elementary Algebra

Question

Convert the following problems from scientific form to standard form. $$ 3.87 \times 10^{5} $$

Step-by-Step Solution

Answer

Question: Convert the scientific notation number $$3.87 \times 10^5$$ to standard form. Answer: $$387,000$$

Step 1: Identify the scientific notation number

We are given the number $$3.87 \times 10^5$$. This number is in scientific notation with a base of 10 raised to the power of 5.

Step 2: Multiply the decimal number by the power of 10

We need to multiply 3.87 by 10^5. This means we have to move the decimal point 5 places to the right while filling any gaps with zeros.

Step 3: Move the decimal point

Moving the decimal point 5 places to the right, we get: $$3.87 \times 10^5 = 387,000$$

Step 4: Write the final answer in standard form

Our final answer, in standard form, is: $$387,000$$

Key Concepts

Standard Form

Standard form is a way of writing numbers that is very straightforward and easy to read. It involves writing numbers out fully with all their digits, without the need for powers or exponents. In the exercise we had the scientific notation number \(3.87 \times 10^5\), which was ultimately expressed in standard form as \(387,000\).

Standard form allows us to see the full number as it is, without any shorthand. This is particularly helpful in understanding just how large or small a number actually is. Imagine having to calculate a large sum, knowing the full number gives you a clear picture of the quantities you're working with.

Transitioning from scientific notation to standard form involves the process of moving decimal points to fit the exact value of the number, which is crucial in fields of science and engineering.

Powers of Ten

Powers of ten are a key component of scientific notation. They represent how many times you need to multiply the number by 10. For instance, in the expression \(3.87 \times 10^5\), \(10^5\) signifies that 3.87 needs to be multiplied by 10 five times.

This is tantamount to adding five zeros to 3.87, thereby transforming it into 387,000. Powers of ten are incredibly useful in mathematical calculations involving very large or very small numbers. By compressing lengthy numbers into a more digestible format, they simplify and accelerate computations.

Every power of ten indicates a shift in magnitude. If the power is positive, like in \(10^5\), this means we are dealing with a large number. Conversely, if the exponent was negative, it would imply a division by ten, useful for very small numbers.

Decimal Point Movement

Decimal point movement is essential when converting between scientific notation and standard form. The decimal in our example \(3.87 \times 10^5\) moved five places to the right because the exponent was 5.

Think of the decimal point as a marker indicating the boundary between the whole number and fractional part of a number. In scientific notation, moving this marker helps account for the multiplicative effect of powers of ten.

Here's how it works in a simple way:

• For each increase in the power of ten, shift the decimal one place to the right.
• If the exponent had been negative, like \(10^{-3}\), you would move the decimal to the left, useful for showing smaller scale numbers.

This movement transforms the number into its full form and helps in perceiving the number's actual size in standard form. It ties the other concepts together, ensuring you fully understand the numeric value being represented.