Problem 61 - College Algebra with Modeling and Visualization

Question

Use a calculator to approximate the expression. Write your result in scientific notation. $$ \frac{8.947 \times 10^{7}}{0.00095}\left(4.5 \times 10^{8}\right) $$

Step-by-Step Solution

Answer

The approximate result is \( 4.24 \times 10^{23} \).

Step 1: Simplify the Expression

First, we need to simplify the given expression. The expression is \( \frac{8.947 \times 10^{7}}{0.00095} \times 4.5 \times 10^{8} \). We can split this into two parts: the division \( \frac{8.947 \times 10^{7}}{0.00095} \) and the multiplication by \( 4.5 \times 10^{8} \).

Step 2: Calculate the Division

Now let's calculate the division part: \( \frac{8.947 \times 10^{7}}{0.00095} \). Use a calculator to find: \( \frac{8.947}{0.00095} = 9417.8947 \). Considering the powers of 10: \( 10^7 / 10^{-4} = 10^{7+4} = 10^{11} \). Thus, \( \frac{8.947 \times 10^{7}}{0.00095} \approx 9417.8947 \times 10^{11} \).

Step 3: Multiply the Result by the Remaining Term

Next, multiply \( 9417.8947 \times 10^{11} \) by \( 4.5 \times 10^{8} \). First, calculate the numeric part: \( 9417.8947 \times 4.5 = 42380.52615 \). Now, for the powers of 10: \( 10^{11} \times 10^{8} = 10^{19} \). Therefore, the expression simplifies to \( 42380.52615 \times 10^{19} \).

Step 4: Convert to Scientific Notation

Now, convert \( 42380.52615 \times 10^{19} \) into proper scientific notation. Move the decimal point four places to the left: \( 4.238052615 \times 10^{23} \). Thus, the final result in scientific notation is approximately \( 4.24 \times 10^{23} \) after rounding to three significant figures.

Key Concepts

Calculator Use

When dealing with complex mathematical expressions, a calculator is an invaluable tool to achieve precision quickly. Here's a simple guide to using a calculator effectively:

• First, identify parts of the expression that require division or multiplication. In our exercise, we have: \( \frac{8.947 \times 10^{7}}{0.00095} \).
• Enter these numbers into the calculator in the order of operations. Begin with the division: type \(8.947\), press the division key, type \(0.00095\), and press equals to get \(9417.8947\).
• Proceed to multiplication: for the next phase, input \(9417.8947 \times 4.5\) to obtain a result.
• For operations involving powers of ten, perform them separately (or keep track with mental arithmetic or the calculator’s exponent function if available).

A practiced use of a calculator speeds up these processes ensuring accuracy without the errors manual computation might bring.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves breaking down expressions into most basic forms. Here, focus on observing grouping symbols and operation orders, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
In our example:

• Look at \(\frac{8.947 \times 10^{7}}{0.00095}\). Solve the division first.
• Handle multiplication by \(4.5 \times 10^{8}\) separately after solving the division.
• Ensure each part is simplified step-by-step before combining results.

Simplifying ensures our expressions are in a manageable form for further computations or transformations into scientific notation.

Powers of Ten

Understanding powers of ten is crucial when simplifying expressions in scientific notation. A power of ten shows how many times to multiply 10 by itself. For example, \(10^{3}\) means \(10 \times 10 \times 10 = 1000\).
In the given exercise, the goal is to keep track of powers of ten throughout calculations.

• In division: \(10^{7} / 10^{-4}\) results in adding exponents: \(10^{7 + 4} = 10^{11}\).
• In multiplication: \(10^{11} \times 10^{8}\) means adding exponents again: \(10^{11+8} = 10^{19}\).

Powers of ten are added during multiplication and subtracted during division, streamlining the process of combining large and small numbers.

Division and Multiplication Steps

The essential steps involved in division and multiplication when solving expressions can simplify complex equations into concrete solutions.

• Division: Start by dividing the coefficients: \(\frac{8.947}{0.00095} = 9417.8947\). Then apply powers of ten as needed, shown by exponent arithmetic such as \(10^{11}\) from \(10^{7} / 10^{-4}\).
• Multiplication: Multiply result from division by the next numerical term: \(9417.8947 \times 4.5\), ensuring to separately handle numeric and exponential components. This yields \(42380.52615\).
• Exponent handling involves adding the powers when terms are in the form of powers of ten: \(10^{11} \times 10^{8}\) becomes \(10^{19}\).

This systematic handling of division then multiplication simplifies complex expressions, preparing them for scientific notation and further use.