Problem 79 - Elementary Algebra

Question

Convert the following problems from scientific form to standard form. $$ 1 \times 10^{9} $$

Step-by-Step Solution

Answer

Question: Convert the given number in scientific notation, \(1 \times 10^9\), to standard form. Answer: 1,000,000,000

Step 1: Identify the power of 10

In this case, the power of 10 is 9, meaning we need to multiply 1 by 10 raised to the power of 9.

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

We need to multiply 1 by 10 raised to the power of 9: $$ 1 \times 10^9 $$

Step 3: Represent the power of 10 in standard form

Now, we need to write 10 raised to the power of 9 as a standard number by multiplying 10 by itself 8 more times: $$ 10^9 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1,000,000,000 $$

Step 4: Write the final result in standard form

Finally, we can write the given number \(1 \times 10^9\) in standard form as: $$ 1 \times 1,000,000,000 = 1,000,000,000 $$ So, the given number in standard form is 1,000,000,000.

Key Concepts

Standard Form

Standard form is a way of writing numbers that are too big or too small to be written conveniently in decimal form. It is especially useful in scientific fields where precise and large quantities are often dealt with. In standard form, a number is expressed as a base number multiplied by a power of 10. This allows for efficient handling and communication of very large or very small numbers, by simplifying their representation.

For example, instead of writing 1,000,000,000, we can express this number in scientific notation as \(1 \times 10^9\), and convert it back to standard form for readability and understanding. This conversion helps to better visualize the scale and magnitude of values involved in scientific computations.

Exponents

An exponent is a mathematical notation that indicates the number of times a number, known as the base, is multiplied by itself. It is written as a small number to the upper right of the base number. In the expression \(10^9\), 10 is the base and 9 is the exponent.

Exponents are powerful tools in mathematics because they provide a shorthand way to express repeated multiplication. This is particularly handy when dealing with large numbers, as they allow us to represent them without writing out long strings of zeros. In our given exercise, the exponent 9 tells us to multiply 10 by itself 9 times, resulting in a substantial number.

Multiplication

Multiplication is a basic arithmetic operation that combines two numbers to produce a third, called the product. It is foundational in mathematics and essential for understanding scientific notation.

In the process of converting scientific notation to standard form, multiplication plays a key role. Specifically, the base number (in this case, 1) is multiplied by 10 raised to the power indicated by the exponent. Through multiplication, we take 1 and perform the operation \(1 \times 10^9\) to find its equivalent value in standard form, resulting in 1,000,000,000.

Powers of 10

Powers of 10 are expressions that show 10 multiplied by itself a certain number of times. These are useful in scientific and mathematical calculations due to their simplicity and the ability to represent large or small numbers efficiently.

When working with powers of 10, each power corresponds to a shift in the decimal place to the right (for positive exponents) or to the left (for negative exponents). In our example, \(10^9\) represents 10 multiplied by itself 9 times, equating to 1,000,000,000. This scalability is particularly advantageous when converting from scientific to standard form, making complex calculations and data easier to understand and handle.