Problem 99 - Algebra A Combined Function

Question

Evaluate each expression using exponential rules. Write each result in standard form. $$ \left(4 \times 10^{-10}\right)\left(7 \times 10^{-9}\right) $$

Step-by-Step Solution

Answer

The result is \(2.8 \times 10^{-18}\).

Step 1: Identify the Components

Recognize that the expression is a product of two numbers in scientific notation: \(4 \times 10^{-10}\) and \(7 \times 10^{-9}\).

Step 2: Multiply the Coefficients

Multiply the coefficients (the numbers before the powers of ten). Compute \(4 \times 7 = 28\).

Step 3: Apply the Exponential Rule for Multiplication

Use the rule \((a^m)(a^n) = a^{m+n}\) for multiplying powers of ten. Here, multiply \(10^{-10}\) and \(10^{-9}\), which gives \(10^{-10 + (-9)} = 10^{-19}\).

Step 4: Combine Results

Combine the results of the coefficient multiplication and the powers of ten: \(28 \times 10^{-19}\).

Step 5: Convert to Standard Form

To express \(28 \times 10^{-19}\) in standard form (one non-zero digit to the left of the decimal), write it as \(2.8 \times 10^{-18}\).

Step 6: Final Result

The expression \((4 \times 10^{-10})(7 \times 10^{-9})\) evaluates to \(2.8 \times 10^{-18}\) in standard form.

Key Concepts

Exponential Rules

Exponential rules are guidelines that help simplify expressions involving powers or exponents, making calculations more manageable. When multiplying two numbers with the same base, an important rule applies: you simply add the exponents. The general formula is

• \( (a^m) \times (a^n) = a^{m+n} \)

This rule is extremely useful when dealing with large or very small numbers expressed as powers of ten, as it allows us to combine these terms efficiently. For instance, in scientific notation, where the base number is always ten, this rule helps us deal with products like \(10^{-10}\) and \(10^{-9}\), simplifying them to \(10^{-19}\) by merely adding the exponents. This keeps calculations simple and avoids handling unwieldy numbers directly.

Multiplying Scientific Notation

Scientific notation is a shorthand way of writing very large or small numbers using powers of ten. Multiplying numbers in scientific notation involves a straightforward two-step process: multiply the coefficients (the number parts), and then apply the exponential rules to the powers of ten. For example, when finding

• \((4 \times 10^{-10})\)
• \((7 \times 10^{-9})\)

Begin by multiplying the coefficients: \(4 \times 7 = 28\). Next, deal with the exponents of the powers of ten using the rule \((10^{-10})(10^{-9}) = 10^{-19}\). The final expression becomes \(28 \times 10^{-19}\). This method ensures accuracy and simplicity when working with scientific notation.

Standard Form

Standard form, also known as scientific notation, is a way to write numbers in the form \(a \times 10^n\) where the coefficient \(a\) has one non-zero digit to the left of the decimal point. This is crucial for simplifying the presentation of very large or small numbers, making them easier to read and work with. To convert the expression \(28 \times 10^{-19}\) into standard form, adjust the coefficient to fulfill this requirement. Divide 28 by 10 to get 2.8 and accordingly, adjust the exponent by adding 1 to \(-19\), giving

• \(2.8 \times 10^{-18}\)

This represents the same value but adheres to the standard form, making it easier to interpret and use in further calculations or comparisons.