Problem 85 - Elementary Algebra Within Reach

Question

Evaluate the expression. $$9.3 \times 10^{6}$$

Step-by-Step Solution

Answer

The evaluated value of the expression \(9.3 \times 10^{6}\) is 9,300,000.

Step 1: Understanding Scientific Notation

In the given expression, the number is in scientific notation format, where \(9.3\) is the coefficient and \(10^{6}\) represents the power of 10. The value of \(10^{6}\) is '1' followed by '6' zeros. We can therefore regard it as 1,000,000.

Step 2: Multiplication

As the final step, multiply the coefficient (9.3) with the value of \(10^{6}\) (which is 1,000,000). When you multiply 9.3 by 1,000,000, you will get the final answer.

Key Concepts

Exponents in Mathematics

Exponents are a fundamental concept in mathematics. They allow us to represent repeated multiplication of a number by itself in a simpler way. For instance, when you see an expression like \(10^{6}\), the exponent is \(6\). This tells you that the base, which is \(10\), is multiplied by itself 6 times:

• \(10 \times 10 \times 10 \times 10 \times 10 \times 10\)

In simple words, an exponent gives you a shorthand way to describe how many times to use a number in a multiplication.
It's essential when dealing with large numbers because it reduces writing and helps in making calculations much easier. In the expression from our exercise, \(10^6\) is an example of how exponents can simplify writing of what otherwise would be a large number.

Multiplying Decimals

When multiplying decimals, it might seem a bit tricky at first, but it follows the same principles as multiplying whole numbers. Here's what to keep in mind:

• Ignore the decimal point and multiply the numbers as if they were whole numbers. For example, multiply 93 by 1,000,000 instead of 9.3 by 1,000,000 initially.
• Count the total number of decimal places in the factors. In our case, 9.3 has 1 decimal place.
• Place the decimal in the result, ensuring it has the same number of decimal places as the total number you've counted. For 9.3 multiplied by 1,000,000, write the answer first as 9,300,000 and then place the decimal point to get 9,300,000.0.

This step-by-step approach makes it easier to track the decimal placement and avoid mistakes. It keeps calculations accurate and ensures you get the correct result.

Powers of Ten

Powers of ten are a tool that can simplify understanding large or small numbers. Each power of ten is a number 1 followed by zeros. The exponent tells you how many zeros to write.

• \(10^1 = 10\) (which is 1 zero)
• \(10^2 = 100\) (which is 2 zeros)
• \(10^3 = 1,000\) (which is 3 zeros)

This pattern continues indefinitely, allowing us to handle even extremely large numbers with simplicity. For \(10^6\), you have a 1 followed by 6 zeros: 1,000,000.
Using powers of ten helps a lot in scientific notation, where large numbers like 9,300,000 can be easily represented as \(9.3 \times 10^6\). This makes working with and communicating about numbers much more efficient, especially in scientific and technical contexts.