Problem 64 - Elementary and Intermediate Algebra

Question

Use scientific notation to perform the calculations. Give all answers in scientific notation and standard notation. \(\left(5.5 \times 10^{-15}\right)\left(2.2 \times 10^{13}\right)\)

Step-by-Step Solution

Answer

The product is \(1.21 \times 10^{-1}\) in scientific notation and 0.121 in standard notation.

Step 1: Rewrite the Problem in Scientific Notation

We have the expression \( (5.5 \times 10^{-15}) \times (2.2 \times 10^{13}) \). This is already in scientific notation, so retain this form for calculations.

Step 2: Multiply the Coefficients

Multiply the decimal numbers, ignoring any powers of ten initially. Calculate: \( 5.5 \times 2.2 = 12.1 \).

Step 3: Add the Exponents

Add the exponents of the powers of ten: \(-15 + 13 = -2\). So, the product of the powers is \(10^{-2}\).

Step 4: Combine the Results

We now combine the coefficient with the power of ten: \(12.1 \times 10^{-2}\).

Step 5: Convert to Standard Notation

To express the number in standard notation, move the decimal point in 12.1 two places to the left (since the exponent is -2): This gives us 0.121.

Key Concepts

Multiplying Powers of Ten

When you multiply numbers in scientific notation, you're often dealing with exponents of ten. In scientific notation, a number is expressed as the product of a coefficient (a number between 1 and 10) and a power of ten. For example, in the number \(5.5 \times 10^{-15}\), \(5.5\) is the coefficient, and \(10^{-15}\) is the power of ten.
To multiply powers of ten, simply add the exponents. This is because multiplying powers of the same base (in this case, ten) results in adding their exponents together. For example:

• \(10^{-15} \times 10^{13} = 10^{(-15+13)} = 10^{-2}\)

This process makes calculations with scientific notation straightforward and less error-prone, especially when dealing with very large or very small numbers. Remember, this rule only applies to powers of the same base. If the bases are different, you must address the calculation differently.

Standard Notation

Standard notation is another way to express numbers, but unlike scientific notation, it does not involve powers of ten explicitly. Scientific notation is particularly useful for very large or very small numbers, but standard notation is often clearer for numbers that are manageable in everyday contexts.
To convert a number from scientific to standard notation, adjust the decimal point according to the exponent of ten in the notation:

• If the exponent is positive, move the decimal to the right.
• If it's negative, move the decimal to the left.

For example, in the expression \(12.1 \times 10^{-2}\):

• The exponent is \(-2\), so move the decimal two places to the left to get 0.121.

This conversion allows you to see a number in a format that is more familiar in everyday reading and computations.

Multiplying Coefficients

When multiplying numbers in scientific notation, you'll begin by multiplying the coefficients, which are the numerical parts of the expression before the power of ten. It's essential to handle these calculations separately from the exponents.
In the original exercise:

• The coefficients are \(5.5\) and \(2.2\).

Multiply them as you would any regular numbers:

• \(5.5 \times 2.2 = 12.1\)

By first calculating the product of the coefficients, you can then pair it with the combined power of ten, which you already calculated by adding exponents. This two-step approach simplifies the process and ensures you manage both numerical values and powers of ten correctly. The final result is then expressed in scientific notation, tying together both the multiplied coefficient and the calculated power of ten.