Multiplication and Division Operations: Videos & Practice Problems

When performing operations with numbers in scientific notation, the process differs for multiplication and division. For multiplication, you multiply the coefficients (the numbers in front of the exponential part) and add the exponents. For example,

$(A\times {10}^x)\times (B\times {10}^y)=(A\times B)\times {10}^{x+y}$

In division, you divide the coefficients and subtract the exponents, such as

$(A\times {10}^x)÷(B\times {10}^y)=(A÷B)\times {10}^{x-y}$

It's essential to remember that the final answer must have the least number of significant figures from the coefficients used in the calculation. Additionally, ensure that the coefficient in the scientific notation is between one and less than ten, adjusting the exponent accordingly if you move the decimal point. For instance,

$10.4822\times {10}^8\to 1.0\times {10}^9$

to maintain proper scientific notation with the correct number of significant figures.

Scientific Notation in Multiplication and Division

When working with numbers in scientific notation, understanding how to multiply and divide them is essential. To multiply two values expressed in scientific notation, you first multiply their coefficients (the numerical parts) and then add their exponents (the powers of ten). For example, if you have \( a \times 10^x \) and \( b \times 10^y \), the result of the multiplication will be:

\[ (a \times b) \times 10^{(x+y)} \]

Conversely, when dividing numbers in scientific notation, you divide the coefficients and subtract the exponents. For instance, if you divide \( a \times 10^x \) by \( b \times 10^y \), the result will be:

\[ \left(\frac{a}{b}\right) \times 10^{(x-y)} \]

It is also important to consider significant figures when performing these operations. When multiplying or dividing, the result should reflect the least number of significant figures present in any of the original numbers. In contrast, when adding or subtracting, the result should have the least number of decimal places.

By applying these principles, you can accurately handle calculations involving scientific notation, ensuring precision in your results. Practice with examples will further solidify your understanding of these concepts.

Scientific Notation Multiplication

When multiplying values in scientific notation, the process involves two main steps: multiplying the coefficients and adding the exponents. For instance, if we have three values: \(2.13 \times 10^5\), \(1.66 \times 10^3\), and \(3.07 \times 10^6\), we first multiply the coefficients:

\(2.13 \times 1.66 \times 3.07 = 10.4822\).

Next, we add the exponents together:

\(5 + 3 + 6 = 14\).

Thus, the preliminary result is \(10.4822 \times 10^{14}\). However, we must consider significant figures. The number of significant figures in a multiplication or division operation is determined by the value with the least significant figures. In this case, \(2.13\) has three significant figures, \(1.66\) has three, and \(3.07\) also has three. Therefore, the least number of significant figures is three.

To express the result in proper scientific notation, the coefficient must be between 1 and 10. Since \(10.4822\) is not in this range, we adjust it:

We move the decimal one place to the left, resulting in \(1.04822\), and increase the exponent by one, leading to \(1.04822 \times 10^{15}\). Rounding this to three significant figures gives us \(1.05 \times 10^{15}\).

When using a calculator, it is crucial to input the numbers in parentheses to ensure accurate calculations. For example, inputting \((2.13 \times 10^5) \times (1.66 \times 10^3) \times (3.07 \times 10^6)\) will yield the correct result. Following these steps will help you accurately multiply numbers in scientific notation.

Scientific Notation Mixed Operations

To solve a mixed operations problem involving multiplication and division, it is essential to follow a systematic approach. Start by identifying the coefficients and exponents in the expression. For instance, if you have coefficients that multiply together to yield 11.8454, you will also need to handle the exponents appropriately. When multiplying terms with exponents, you add the exponents together. In this case, if the exponents are 8 and -1, their sum would be 7.

Next, when dividing, remember that this operation translates to subtracting the exponents. For example, dividing \(10^7\) by \(10^{11}\) results in \(10^{7-11} = 10^{-4}\). However, the coefficient must be expressed in scientific notation, which requires it to be a number between 1 and 10. To adjust the coefficient from 11.8454 to 1.18454, you move the decimal point one place to the left, which necessitates increasing the exponent from -4 to -3.

Finally, it is crucial to consider significant figures when presenting your final answer. The number of significant figures in the result should match the term with the least significant figures in the calculation. If all coefficients have 3 significant figures, your final answer should also reflect this. Thus, the properly formatted answer would be \(1.18 \times 10^{-3}\).

Here’s what students ask on this topic:

How do you multiply numbers in scientific notation?

To multiply numbers in scientific notation, you multiply the coefficients (the numbers in front of the exponential part) and add the exponents. For example, if you have (A × 10^x) × (B × 10^y), you multiply A and B to get a new coefficient, and add x and y to get the new exponent. The result is (A × B) × 10^x+y. Ensure the final coefficient is between 1 and less than 10, adjusting the exponent if necessary.

How do you divide numbers in scientific notation?

To divide numbers in scientific notation, you divide the coefficients and subtract the exponents. For example, if you have (A × 10^x) ÷ (B × 10^y), you divide A by B to get a new coefficient, and subtract y from x to get the new exponent. The result is (A ÷ B) × 10^x-y. Ensure the final coefficient is between 1 and less than 10, adjusting the exponent if necessary.

What are the rules for significant figures in multiplication and division?

When multiplying or dividing numbers, the result should have the same number of significant figures as the value with the least number of significant figures used in the calculation. For example, if you multiply 2.13 (3 significant figures) by 1.6 (2 significant figures), the result should be rounded to 2 significant figures.

How do you handle mixed operations with scientific notation?

For mixed operations involving multiplication and division in scientific notation, follow the order of operations: perform multiplication and division from left to right. Multiply or divide the coefficients and handle the exponents accordingly (add for multiplication, subtract for division). Ensure the final coefficient is between 1 and less than 10, adjusting the exponent if necessary, and round the result to the correct number of significant figures.

Why is it important to use parentheses when entering scientific notation into a calculator?

Using parentheses when entering scientific notation into a calculator ensures that the operations are performed correctly. For example, when multiplying (2.13 × 10⁴) × (1.66 × 10³), enter it as (2.13E4) * (1.66E3) to avoid errors. Parentheses help the calculator interpret the scientific notation correctly and provide the accurate result.