Question

When powers of 10 are multiplied together, the powers are added together. For example, \(10^{2}\) \(10^{3}=10^{\frac{f+3}{3}}=10^{5} .\) With this in mind, can you evaluate \(\left(4.506 \times 10^{4}\right) \times\left(1.003 \times 10^{2}\right)\) without entering scientific notation into your calculator?

Short answer

Approximately \(4.51 \times 10^{6}\).

Step-by-step solution

Identify the Problem

We need to multiply two numbers expressed in scientific notation: \(4.506 \times 10^{4}\) and \(1.003 \times 10^{2}\).

Separate Multiplication Components

Express the multiplication as the product of the decimal parts and the powers of ten: \((4.506 \times 1.003) \times (10^{4} \times 10^{2})\).

Multiply the Decimal Parts

Multiply the decimal numbers: \(4.506 \times 1.003\). Approximating this multiplication gives \(4.506 + 0.004506 \approx 4.510506\).

Multiply the Powers of Ten

According to the rule, add the exponents: \(10^{4} \times 10^{2} = 10^{4+2} = 10^{6}\).

Combine the Results

Combine the decimal result from the product of the decimals with the power of ten: \(4.510506 \times 10^{6}\).

Approximate the Final Result

For simplicity, and considering significant figures, approximate \(4.510506 \approx 4.51\). Thus, the final result is approximately \(4.51 \times 10^{6}\).

Key concepts

Multiplication of Powers

When you multiply powers of the same base, you're actually performing a mathematical operation that involves adding exponents. The rule here is straightforward: add the exponents together.

The formula is: if you have two expressions like \(a^m \times a^n\), the result is \(a^{m+n}\). For example:

• \(10^2 \times 10^3\) becomes \(10^{2+3} = 10^5\)

This property makes multiplying large numbers much simpler, especially when they appear as powers of ten in scientific notation. The process not only saves time but also reduces the chances of error when calculating with large exponents.

Exponents Rules

Exponents are small numbers placed to the top right of a base number, indicating how many times to multiply the base by itself. Understanding the rules of exponents can make complex calculations very manageable. Here are the core rules you'll encounter:

• Product of Powers Rule: Add exponents when multiplying the same base, as in \(x^m \times x^n = x^{m+n}\).
• Power of a Power Rule: Multiply exponents when raising a power to another power, such as \((x^m)^n = x^{m\times n}\).
• Power of a Product Rule: Apply the exponent to all components inside the parentheses: \((xy)^n = x^n \times y^n\).

These rules simplify expressions involving exponents and are essential for solving problems like the multiplication of powers.

Multiplication of Decimals

Multiplying decimals is similar to multiplying whole numbers, with a few extra considerations needed for precision. When multiplying two decimals:

• Ignore the decimal points and multiply the numbers as if they were whole.
• Count the total number of decimal places from both numbers.
• Put the decimal back in the product, ensuring you have the same number of decimal places as the sum of the original decimal places.

For instance, when multiplying 4.506 and 1.003, count a total of six decimal places (three from each number). After multiplication, the final step would be adjusting the result to accommodate these decimal places. This might involve rounding based on significant figures, especially in scientific notation contexts, ensuring your answer is both precise and concise.