Problem 82 - College Algebra

Question

Perform the indicated operation and express the answer in decimal notation. $$ \left(5 \times 10^{2}\right)\left(4 \times 10^{4}\right) $$

Step-by-Step Solution

Answer

The result of the operation in decimal notation is 20000000.

Step 1: Separation

Separate the decimal and base-10 parts of both numbers which will give: (5 * 4) and \((10^2 * 10^4)\).

Step 2: Perform Multiplication

Perform multiplication separately for decimal and exponential parts. So, you get 20 and \(10^{2+4}\). According to the rules of exponents, when multiplying, the exponents are added when they have the same base.

Step 3: Simplify the Result

Simplify the result, which will yield \(20 * 10^6\).

Step 4: Convert to Decimal Notation

Finally, convert the multiplication result into decimal notation, which gives 20000000.

Key Concepts

Exponents and Powers

Exponents and powers represent a way to express repeated multiplication of the same number. When a number is raised to a power, the number is known as the base, and the exponent tells us how many times to multiply the base by itself. For instance, if we have the expression \(2^3\), it means that we should multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\).

This principle is particularly handy when dealing with large numbers in scientific notation, where expressions like \(10^2\) mean 10 multiplied by itself twice, which results in 100. Understanding how exponents work is crucial in many areas of mathematics, including algebra and calculus.

Decimal Notation

Decimal notation is a method of writing numbers that uses the base 10 system. It is composed of a whole number part, a decimal point, and a fractional part. The position of each digit in relation to the decimal point signifies its value or place value. For example, \(31.45\) represents 31 whole units plus 45 hundredths.

When converting scientific notation to decimal notation, you expand the number by the power of ten indicated. In the given exercise, converting the scientific notation of \(20 \times 10^6\) into decimal notation results in the expansion of 20 by six places to the right, yielding 20,000,000. Accurate conversion between these notations is essential for correctly interpreting numbers in scientific and real-world contexts.

Multiplying Powers of Ten

Multiplying powers of ten is significantly simplified by one of the rules of exponents: when you multiply numbers that have the same base, you add their exponents. In the world of base ten, which our numeric system relies on, this rule is incredibly useful. If we have \(10^2\) and \(10^3\), multiplying them gives us \(10^{2+3} = 10^5\), rather than calculating the product of 100 and 1000, which can be cumbersome without this exponent rule.

Understanding this process is important not only for solving math problems but also for recognizing patterns in data, such as scientific measurements and financial calculations that often involve very large or very small numbers.

Simplifying Exponential Expressions

Simplifying exponential expressions involves applying the rules of exponents to make expressions more manageable and easier to understand. Apart from multiplying powers of ten, there's a wide range of operations that can be simplified, including dividing expressions with the same base (which requires you to subtract the exponents) and raising a power to another power (where you multiply the exponents).

For example, in our textbook exercise, the expression \(20 \times 10^6\) is already simplified, but if we had an expression like \((10^4)^3\), simplifying it would result in \(10^{4 \times 3} = 10^{12}\). Efficiently simplifying expressions is a cornerstone in algebra and helps with solving complex problems by breaking them down into simpler steps.