Problem 67 - Elementary Algebra

Question

Perform the following operations. $$ \left(9 \times 10^{-5}\right)\left(1 \times 10^{-11}\right) $$

Step-by-Step Solution

Answer

Question: Multiply the given numbers in scientific notation: $(9 \times 10^{-5})\left(1 \times 10^{-11}\right)$ Answer: $9 \times 10^{-16}$

Step 1: Multiply the coefficients

Multiply the coefficients (numbers without the exponent parts): $$ 9 \times 1 = 9 $$

Step 2: Add the exponents

Add the exponents of the numbers: $$ -5 + (-11) = -16 $$

Step 3: Combine the result

Combine the result of Step 1 and Step 2 to get the final answer in scientific notation: $$ 9 \times 10^{-16} $$ So, $$ \left(9 \times 10^{-5}\right)\left(1 \times 10^{-11}\right) = 9 \times 10^{-16} $$

Key Concepts

Multiplying Exponents

When you encounter exponents in mathematical operations, particularly in scientific notation, remember that exponents indicate how many times a number is multiplied by itself. In the case of multiplying numbers in scientific notation, you'll see exponents working together. The numbers in scientific notation are expressed in the form of

• coefficient \( \times 10^{exponent} \)

When multiplying these terms, focus on two main steps:
1. **Multiply the Coefficients**: This simply means taking the given coefficients (the numbers before the exponent part) and multiplying them like you would in basic arithmetic. For example, if you have 9 and 1 as coefficients, multiply them to get 9.
2. **Add the Exponents**: When multiplying terms with the same base in this format—in our case 10—you add their exponents. So, for \(10^{-5}\) and \(10^{-11}\), you add -5 and -11 to get -16.

Now, combine the results from these two operations to get your final answer in scientific notation.

Coefficients

Coefficients are the numbers that lead in a scientific notation expression, and they play an important role when performing multiplication. Understanding coefficients is straightforward: they are simply the base numbers that you will work with before dealing with exponents.

• In scientific notation, a coefficient is placed in front of the base 10 and is multiplied as part of solving the expression.
• For example, in the expression \(9 \times 10^{-5}\), 9 is the coefficient.

When multiplying coefficients:

• Simply multiply the numbers as you would with usual arithmetic.
• In the exercise given, multiplying the coefficients 9 and 1 results in 9.

Once coefficients are multiplied, combine them with the calculated result of the exponents to finalize your scientific notation solution.

Elementary Algebra

Elementary algebra involves basic algebraic principles, including operations with numbers and variables. When working with scientific notation in algebraic expressions, understanding is built upon understanding simple arithmetic but with added rules:

• **Multiplication and Division**: These operations require specific rules for both coefficients and exponents, especially when in scientific notation.
• **Addition and Subtraction**: Generally, this involves making sure exponents in scientific notation match before calculating.

In the case of the given problem, we're focused on multiplication using scientific notation, where the main tasks are to:

• Multiply the coefficients directly as normal numbers.
• Apply the rule of summing exponents when they share the same base, which is typical in algebraic manipulation.

Elementary algebra helps you sensibly combine these operations, reflecting the fundamental skills that form the backbone of solving scientific notation problems with confidence.