Question

Given the number \(6.39 \times 10^{7},\) would you move the decimal point to the left or to the right to rewrite the number in decimal form?

Short answer

The decimal point should be moved to the right to rewrite the number in decimal form.

Step-by-step solution

Identify exponent of 10

In the scientific notation, the exponent of 10 is 7. This signifies that the decimal point will be moved 7 places.

Determine direction of movement

If the exponent of 10 is positive, the decimal point is moved to the right, if the exponent is negative, the decimal point is moved to the left. In our case, since the exponent is positive (7), the decimal point should be moved to the right.

Move the decimal point

Move the decimal point in \(6.39 \) seven places to the right. If there are not enough places after the decimal, pad with zeros.

Rewrite the number

After moving the decimal point, rewrite the number in decimal form. This will be the solution.

Key concepts

Exponents

Understanding exponents is essential when working with scientific notation. An exponent, sometimes called a 'power,' tells us how many times to multiply a base number by itself. For example, the exponent in the number \(10^7\) indicates that we should multiply 10 by itself 7 times: \(10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10\). This is equivalent to writing out a one followed by seven zeros in decimal form, or 10,000,000.

In scientific notation, a large number is often expressed as a product of a number between 1 and 10, and a power of 10. The exponent reflects how many places the decimal point can shift to convert the number into a whole number or a decimal, thus drastically simplifying the expression of very large or very small numbers. When an exponent is positive, it represents the magnitude of large numbers, as seen with \(6.39 \times 10^{7}\) in our given example.

Decimal Form Conversion

Converting numbers from scientific notation to decimal form involves understanding the role of the exponent. As we've seen, the exponent dictates the number of places the decimal point will move. If the exponent on the 10 is positive, you'll move the decimal to the right; if it's negative, you'll move it to the left. This simple maneuver adjusts the number's scale without altering its value.

Take the number \(6.39 \times 10^{7}\) from our exercise. To convert this to decimal form, we move the decimal 7 places to the right, because the exponent is 7. Without enough digits to move the decimal the full 7 places, we pad with zeros, resulting in the long-form number 63,900,000. Notice how this conversion process maintains the value of the number but adjusts its format to one that is more familiar and easier to read.

Place Value

Place value in mathematics determines the value of a digit in a number, based on its position. In standard decimal notation, each position to the left of the decimal point represents an increasingly large value, multiplying by 10 each step of the way. This means that each digit is ten times more significant as you move from right to left across a number.

The place values to the left of the decimal in standard notation start with the 'ones' place (the first digit to the left), followed by the 'tens' place, then 'hundreds', and so on. For example, in the number 63,900,000, which we derived from moving the decimal point of \(6.39 \times 10^{7}\), the '6' is in the ten million place, while the '3' is in the million place. Each zero to the right of '3' occupies a sequential place value from hundreds of thousands down to ones, signifying a total absence of value in those positions for this particular number. This hierarchy of place value helps us comprehend the size and scale of numbers quickly, which is especially useful for very large numbers like those often expressed in scientific notation.