Question

Write each number in scientific notation. Perform the operation and write your answer in scientific notation.(a) \(48,000,000,000(250,000,000)\) (b) \(\frac{0.000000012}{0.0000064}\)

Short answer

The results are: (a) \(1.2 \times 10^{19}\) and (b) \(1.875 \times 10^{-3}\).

Step-by-step solution

Convert the numbers to scientific notation

(a) \(48,000,000,000\) can be written as \(4.8 \times 10^{10}\) and \(250,000,000\) can be written as \(2.5 \times 10^{8}\). (b) \(0.000000012\) can be written as \(1.2 \times 10^{-8}\) and \(0.0000064\) can be written as \(6.4 \times 10^{-6}\).

Perform the operations in scientific notation

(a) Multiply \(4.8 \times 10^{10}\) and \(2.5 \times 10^{8}\) to obtain \(12 \times 10^{18}\). (b) Divide \(1.2 \times 10^{-8}\) by \(6.4 \times 10^{-6}\) to obtain \(0.1875 \times 10^{-2}\).

Simplify the solutions

(a) Simplify \(12 \times 10^{18}\) to get \(1.2 \times 10^{19}\). (b) Simplify \(0.1875 \times 10^{-2}\) to get \(1.875 \times 10^{-3}\).

Key concepts

Multiplication in Scientific Notation

When multiplying numbers in scientific notation, the process involves working with both the coefficients and the exponents separately. This makes calculations simpler and saves time.
Begin by multiplying the decimal parts of the numbers. For instance, if you have \(4.8 \times 10^{10} \times 2.5 \times 10^{8}\), first multiply \(4.8\) by \(2.5\) which gives \(12\).
Then, for the powers of ten, you'll add the exponents together. In this example, adding \(10\) and \(8\) gives you \(18\). So, the result before simplification becomes \(12 \times 10^{18}\).

• Separate the operation into parts.
• Multiply the coefficients.
• Add the exponents.

Remember that if the resultant coefficient isn't between 1 and 10, further simplification is needed.

Division in Scientific Notation

Division in scientific notation allows you to handle incredibly small or large numbers with ease. The steps involved are straightforward.
First, divide the coefficients just like regular numbers. For example, let's divide \(1.2 \times 10^{-8}\) by \(6.4 \times 10^{-6}\). Dividing \(1.2\) by \(6.4\) results in approximately \(0.1875\).
Then, subtract the exponent of the divisor from the exponent of the dividend. In this case, subtracting \(-6\) from \(-8\) gives \(-2\). The preliminary result of the division is thus \(0.1875 \times 10^{-2}\).

• Divide the coefficients.
• Subtract the exponents.
• Adjust if needed to fit scientific notation's requirements.

If the resulting coefficient isn't between 1 and 10, you'll need one more step of simplification.

Understanding Exponents in Scientific Notation

Exponents are a vital part of scientific notation, enabling large or tiny numbers to be easily expressed and manipulated.
In scientific notation, each number is expressed as a product of a number (the coefficient) and ten raised to an exponent.
This exponent indicates how many places the decimal point has moved, either to the left for negative exponents or to the right for positive exponents.
For multiplication, you add the exponents of the 10s. For division, you subtract the denominators exponent from the numerator's.

• Positive exponents represent large numbers.
• Negative exponents detail very small numbers.
• Add exponents when multiplying; subtract when dividing.

Working with exponents correctly ensures that operations in scientific notation yield accurate results.

Simplification in Scientific Notation

After performing operations in scientific notation, the results often need simplification for proper expression.
For example, if you arrive at an answer like \(12 \times 10^{18}\), the coefficient \(12\) is not between 1 and 10. Simplify this by transforming it into \(1.2 \times 10^{19}\) by adjusting both the coefficient and the exponent.
Similarly, a result of \(0.1875 \times 10^{-2}\) needs the coefficient \(0.1875\) adjusted to \(1.875\) and the exponent becomes \(-3\) thereby making it \(1.875 \times 10^{-3}\).

• Check if the coefficient is between 1 and 10.
• Adjust coefficient and exponent accordingly.

Simplification ensures the results maintain the proper scientific notation format.