Problem 65 - College Algebra with Modeling and Visualization

Question

Use a calculator to approximate the expression. Write your result in scientific notation. $$ \left(8.5 \times 10^{-5}\right)\left(-9.5 \times 10^{7}\right)^{2} $$

Step-by-Step Solution

Answer

The result is approximately \(7.67125 \times 10^{11}\).

Step 1: Simplify Inside the Parentheses

First, calculate the expression inside the parentheses: \((-9.5 \times 10^{7})^{2}\). This involves squaring both the coefficient and the exponent.\(-9.5^2 = 90.25\) and \((10^{7})^2 = 10^{14}\). So, \((-9.5 \times 10^{7})^2 = 90.25 \times 10^{14}\).

Step 2: Multiply the Coefficients

Next, multiply the coefficients from the two parts of the expression: \(8.5\) and \(90.25\).\\[8.5 \times 90.25 = 767.125\]

Step 3: Multiply the Powers of 10

Now, multiply the powers of 10: \(10^{-5} \times 10^{14}\). According to the laws of exponents, add the exponents: \(-5 + 14 = 9\). So this becomes \(10^{9}\).

Step 4: Combine the Results

Combine the results from the multiplication steps: \(767.125 \times 10^{9}\).

Step 5: Convert to Scientific Notation

Express the final result in scientific notation. Move the decimal one place left in \(767.125\) to get \(7.67125\), and adjust the power of 10 from \(10^9\) to \(10^{10}\) to maintain the value. The final expression in scientific notation is \(7.67125 \times 10^{11}\).

Key Concepts

Exponent Laws

Exponent laws are fundamental when working with scientific notation and calculations involving powers. Understanding these can simplify seemingly complex mathematical expressions. Essentially, the key exponent laws to remember are:

• Product of Powers Rule: When multiplying two expressions with the same base, simply add their exponents. For example, if you have \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: If you have an exponent raised to another power, multiply the exponents. This is seen as \((a^m)^n = a^{m\cdot n}\).
• Negative Exponent Rule: A negative exponent \(a^{-n}\) indicates a reciprocal, meaning \(\frac{1}{a^n}\).

In our exercise, the Power of a Power Rule was used to square \((-9.5 \times 10^{7})\). Squaring resulted in \(90.25 \times 10^{14}\), achieved by squaring the coefficient and multiplying the exponent \(7\) by \(2\). Moreover, the Product of Powers Rule was applied when multiplying \(10^{-5}\) and \(10^{14}\), which resulted in \(10^{9}\). These rules make simplifying computations straightforward.

Calculator Usage

Using a calculator effectively is essential, especially for operations involving high precision and large numbers, typically found in scientific notation. Calculators today are equipped to handle scientific notation directly, which simplifies work significantly.
When using a calculator:

• Locate the scientific notation functionality, often denoted by "EE" or "EXP" button.
• Input numbers in segments, such as \(8.5\) or \(90.25\), and use the scientific notation button to enter exponents like \(10^{-5}\) or \(10^{14}\).
• Practice multiplying and checking large coefficients to ensure accuracy.

For the given problem, first enter \(-9.5\) and square it, followed by entering \(10^7\) and squaring that as well. Then, use the result to proceed with multiplying with \8.5\ and \(10^{-5}\). Familiarity with these calculator functions will make exprssions like \( (-9.5 \times 10^{7})^2 \) easier to calculate, leading to faster and more accurate results.

Multiplying Powers of Ten

Understanding how to multiply powers of ten is crucial in scientific notation, especially for simplifying expressions and achieving correct results. In scientific notation, multiplication of powers of ten can be streamlined using exponents, thanks to the exponent laws.
Here’s how to manage it effectively:

• Recall that multiplying two powers of ten, like \(10^a\) and \(10^b\), involves adding their exponents: \(10^{a+b}\).
• This operation simplifies the expression to a single power of ten, making it clearer and easier to handle.
• Be mindful of negative exponents, as they affect the base and therefore the result.

In our exercise, to multiply \(10^{-5}\) and \(10^{14}\), add the exponents \(-5\) and \(14\) to arrive at \(10^{9}\). Keeping track of exponent signs (positive or negative) is essential to avoid errors. Practicing this operation can enhance speed and accuracy in complex calculations, making it less daunting and second nature over time.