Problem 92 - Algebra and Trigonometry

Question

Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. $$ \left(5.1 \times 10^{-8}\right)\left(3 \times 10^{-4}\right) $$

Step-by-Step Solution

Answer

The result of the multiplication in scientific notation, to two decimal places, is \(1.53 \times 10^{-10}\)

Step 1: Multiplication of Decimal Parts

First, multiply the decimal numbers 5.1 and 3 which results in 15.3

Step 2: Adding the Powers of 10

Next, add the powers of 10, -8 and -4, which results in -12.

Step 3: Formulating the Answer

Combine the results from the previous two steps into the form \(a \times 10^{n}\), which gives the result \((15.30 \times 10^{-12})\). However this is not a correct scientific notation, as the decimal part should be 1 or greater than 1 but less than 10. To correct this, move the decimal point in 15.30 two places to the left, reducing the exponent by 2, yielding an answer of \(1.53 \times 10^{-10}\).

Key Concepts

Multiplication of Decimals

Multiplying decimals is an essential skill in mathematics and is especially useful in scientific notation. When you multiply two decimal numbers, you do so without initially considering the decimal points. Take the numbers 5.1 and 3 from the exercise. Multiply them just as if they were whole numbers:

• 5.1 becomes 51 (ignoring the decimal point)
• 51 multiplied by 3 gives you 153

To place the decimal correctly in the product, you need to count the total number of decimal places in the numbers being multiplied. In our case:

• 5.1 has one decimal place
• 3 is a whole number but can be seen as having zero decimal places

Thus, the product 153 will have one decimal place, turning it into 15.3. Breaking down multiplication this way makes it easier to handle without making errors in decimal placement.

Exponent Rules

Exponent rules are fundamental when working with scientific notation. They simplify handling large or small numbers, like those in the given exercise.When multiplying exponents with the same base, add their powers. This rule makes it easy to work with powers of 10 in scientific notation. In the problem:

• We have the exponents: \(-8\) and \(-4\).
• Adding these gives \(-8 + (-4) = -12\).

This process reduces the complexity of the multiplication operation significantly. Remember, these rules apply because of the properties of exponents, where multiplying like bases allows their exponents to be added: \[a^m \times a^n = a^{m+n}\] This same addition rule is one of the core strengths of using scientific notation, enabling compact and efficient number management.

Rounding in Scientific Notation

Scientific notation helps in expressing very large or very small numbers in a concise form. When writing a number in scientific notation, the decimal part (known as the coefficient) should be between 1 and 10. If needed, rounding the decimal part to a specified number of places is often required for precision. In our problem:

• The multiplication gave 15.3 \( \times 10^{-12} \)
• The coefficient 15.3 needs to be between 1 and 10.

To fix this, move the decimal point two places left, making 15.3 into 1.53. This also affects the exponent. When you move the decimal point left, increase the exponent by the number of moves. Here,

• Moving two places left means increasing the exponent by 2.
• The exponent changes from \(-12\) to \(-10\)

This adjustment keeps the expression precise and within proper scientific notation form, ensuring it is easy to read and understand, while also highlighting significant figures.