Problem 13 - Elementary Algebra

Question

Convert the following numbers to standard form. $$ 9.25 \times 10^{2} $$

Step-by-Step Solution

Answer

Question: Convert the scientific notation $$9.25 \times 10^{2}$$ into standard form. Answer: The standard form of the given scientific notation is $$925$$.

Step 1: Identify the Base and Exponent

The number is given in scientific notation as: $$9.25 \times 10^{2}$$ Here, $$9.25$$ is the base, and $$2$$ is the exponent.

Step 2: Multiply by 10 raised to the power of the exponent

Now, we need to multiply the base by 10 raised to the power of the exponent. In this case, multiply $$9.25$$ by $$10^2$$: $$9.25 \times (10^2)$$

Step 3: Calculate using Exponent Rule

The exponent rule states that $$10^2 = 10 \times 10 = 100$$ $$9.25 \times 10^2 = 9.25 \times 100$$

Step 4: Multiply Base by 100

Now, multiply $$9.25$$ by $$100$$: $$9.25 \times 100 = 925$$

Step 5: Write the Result in Standard Form

So, the number is equal to $$925$$ in standard form.

Key Concepts

Scientific Notation

Scientific notation is a method used to express very large or small numbers in a compact form, making them easier to work with and understand. It is represented as a number between 1 and 10 multiplied by a power of 10. This method is particularly useful in science and engineering, where such numbers frequently occur. For example, the number \(9.25 \times 10^2\) is in scientific notation. Here, 9.25 is the base number in scientific notation, which is always a value between 1 and 10. The base is then multiplied by a power of 10; in this instance, the power is 2, meaning the base will be multiplied by 100. To convert a number from scientific notation to standard form, one needs to multiply the base by 10 raised to the given power.

Exponent Rules

Exponent rules are fundamental guidelines that simplify calculations involving powers of numbers. They provide a way to manipulate exponents when multiplying or dividing numbers. In the context of the exercise, we specifically focus on the rules for powers of ten.

• The power of ten: \(10^n\) means that 10 is multiplied by itself \(n\) times.
• Multiplication Rule: When multiplying, add the exponents if the bases are the same, such as \(10^a \times 10^b = 10^{a+b}\).
• Division Rule: Subtract the exponents when dividing like bases, e.g., \(10^a / 10^b = 10^{a-b}\).

Using these rules helps streamline operations like \(9.25 \times 10^2\), where the application of exponent rules confirms that \(10^2 = 100\), simplifying the multiplication process for converting the number to its standard form.

Multiplication by Powers of 10

Understanding multiplication by powers of 10 is crucial for efficiently converting numbers from scientific notation to standard form. It involves very straightforward operations, especially when the base number is multiplied by 10 raised to an exponent.Given the number \(9.25 \times 10^2\), the operation essentially means multiplying 9.25 by 100, since \(10^2 = 100\). Multiplying by powers of 10 is simplified by shifting the decimal point to the right by the number of places equal to the exponent. For instance, multiplying 9.25 by 100 moves the decimal two places to the right, resulting in 925, which is the standard form of the number.These operations are convenient because they use the properties of the base-ten numeral system, minimizing errors and making calculations faster, especially with large numbers.