Question

In Problems 129-136, write each number as a decimal. \(6.15 \times 10^{4}\)

Short answer

61500

Step-by-step solution

Understand Scientific Notation

Identify that the number is given in scientific notation, which is written as a product of a decimal number and a power of 10.

Identify the Components

Recognize the two components: the decimal number is 6.15 and the exponent is 4.

Move the Decimal Point

Move the decimal point in 6.15 four places to the right because the exponent is positive.

Write the Result

After moving the decimal point, the number becomes 61500.

Key concepts

Decimal Conversion

When converting a number from scientific notation to a standard decimal representation, understanding the basic idea is essential. Scientific notation is a way to express very large or very small numbers neatly, using a base number and an exponent of 10. For example, in the expression 6.15 × 10^4, 6.15 is the base number, and 4 is the exponent.
To convert to decimal format, you need to move the decimal point to the right or left, depending on the exponent. Since the exponent is 4 in this case, you move the decimal point four places to the right. This changes 6.15 to 61500.
Remember, a positive exponent means shifting the decimal point to the right, making the number larger. A negative exponent shifts the decimal point to the left, making the number smaller.

Exponents

Exponents are a fundamental concept in mathematics that indicates how many times a number (the base) is multiplied by itself. In the notation 10^4, 10 is the base and 4 is the exponent. This means 10 multiplied by itself 4 times: 10 × 10 × 10 × 10, which equals 10000.
When dealing with scientific notation, exponents of 10 are particularly useful because they simplify the process of scaling numbers. They help in expressing both very large and very small numbers in a condensed format. For instance, 10^4 is a way to represent 10000 without writing out all the zeros. So, when you see a positive exponent, you know you're dealing with a large number.

Place Value

Place value refers to the numerical value that a digit has by virtue of its position in a number. Each position in a number represents a different power of 10. For example, in the number 61500, the place values are:

• 6 is in the ten-thousands place, representing 60000.
• 1 is in the thousands place, representing 1000.
• 5 is in the hundreds place, representing 500.
• 0 is in the tens place, representing 0.
• 0 is in the ones place, representing 0.

Understanding place value is crucial when converting from scientific notation to decimal form. Moving the decimal point to the right or left effectively changes the place values of the digits, resulting in either a larger or smaller number. So, in 6.15 × 10^4, moving the decimal four places to the right turns 6.15 into 61500.