Problem 14 - Algebra 1: Concepts and Skills

Question

Perform the indicated operation. Write the result in scientific notation. $$ \left(5 \times 10^{6}\right)\left(6 \times 10^{-2}\right) $$

Step-by-Step Solution

Answer

The result of the multiplication \( \left(5 \times 10^{6}\right)\left(6 \times 10^{-2}\right) \) in scientific notation is \(3 \times 10^{5}\).

Step 1: Understand the structure of scientific notation

Recall that a number in scientific notation is of the form \(a \times 10^{n}\), where \(1 \leq |a| < 10\) and \(n\) is an integer. In the given question, the numbers \((5 \times 10^{6})\) and \((6 \times 10^{-2})\) are in this form.

Step 2: Multiply the coefficients and the powers of ten separately

First, multiply the coefficients, which are the numbers outside the powers of ten. So, \(5 \times 6 = 30\). Next, multiply the powers of ten together: \(10^{6} \times 10^{-2}\).

Step 3: Calculate the multiplication of the powers of ten

Recall from the laws of exponents that when you multiply with the same base, you add the exponents. So, \(10^{6} \times 10^{-2} = 10^{6+(-2)} = 10^{4}\).

Step 4: Combine the results

Combine the results of the coefficient multiplication (which was 30) and the powers of ten multiplication (which was \(10^{4}\)). You get \(30 \times 10^{4}\). However, note that this is not in standard scientific notation form as the coefficient 30 is not between 1 and 10. To convert this into correct scientific notation, we rewrite 30 as \(3 \times 10\).

Step 5: Adjust to correct scientific notation

Rewrite the result from Step 4 in correct scientific notation. We get \(3 \times 10 \times 10^{4} = 3 \times 10^{1+4} = 3 \times 10^{5}\). This is the final answer.

Key Concepts

Laws of Exponents

Understanding the laws of exponents is essential for simplifying expressions with powers and performing algebraic operations that involve exponents. An exponent indicates how many times a number, called the base, is multiplied by itself. For instance, in the expression \(2^3\), the base is 2, and the exponent is 3, meaning that 2 is multiplied by itself 3 times: \(2 \times 2 \times 2\).

When multiplying powers with the same base, like \(10^{6} \) and \(10^{-2}\), we use one of the main laws of exponents: we keep the base the same and add the exponents together. This means that \(10^{6} \times 10^{-2} = 10^{6+(-2)} = 10^{4}\). Similarly, if you were to divide powers with the same base, you would subtract the exponents.

Another important rule is that any base with an exponent of 0 is equal to 1 (except for 0 itself). Therefore, \(10^{0} = 1\). These rules enable us to simplify expressions and perform operations much faster and are crucial when working with scientific notation, where powers of ten are frequently used.

Multiplying Powers of Ten

A common operation in mathematics, especially when dealing with scientific notation, is multiplying powers of ten. Since the base is the same (10), this operation is straightforward because of the laws of exponents. In the exercise given, we have the multiplication of \(10^{6}\) and \(10^{-2}\), which demonstrates the application of this rule.

As explained before, to multiply these, you simply add their exponents: \(10^{6} \times 10^{-2} = 10^{6+(-2)} = 10^{4}\). The base remains unchanged (still 10) and the exponents are added, showing just how quickly powers of ten can be combined. This operation is even more efficient than performing the actual multiplication of individual tens. Keep in mind that the resulting exponent could also be negative if subtracting the exponents results in a value below zero, which indicates a fraction rather than a whole number.

Algebraic Operations

Scientific notation is particularly useful in algebra when working with very large or very small numbers. It allows for easier multiplication and division by separating the number into two parts: a coefficient and a power of ten. An essential part of algebraic operations in scientific notation is maintaining the integrity of the notation, ensuring the coefficient stays between 1 and 10.

In our example, we initially multiplied the coefficients (5 and 6) and then multiplied the powers of ten separately. This separation into simpler parts aligns perfectly with the principles of algebra, which often involves breaking down complex problems into more manageable steps. After adjusting the coefficient, we ensured it was within the correct range for scientific notation, which also demonstrates the flexibility of algebraic manipulation.

Performing algebraic operations correctly, whether it is addition, subtraction, multiplication, or division, often depends on understanding the properties of the numbers and operations involved. The simplification steps are fundamental to not only scientific notation but also to broader algebraic problem-solving.