Problem 45 - College Algebra with Modeling and Visualization

Question

Write the number in standard form. $$ 1.567 \times 10^{2} $$

Step-by-Step Solution

Answer

156.7

Step 1: Understanding Standard Form

Standard form for numbers (also known as scientific notation) involves expressing the number as a product of a number between 1 and 10 and an integer power of 10. Here we have the number already in this form: \(1.567 \times 10^2\).

Step 2: Calculate the Power of 10

To convert the scientific notation \(1.567 \times 10^2\) into standard form, you apply the power of 10. Here, \(10^2\) means moving the decimal point 2 places to the right because the exponent is positive.

Step 3: Perform the Calculation

Starting with the number 1.567, move the decimal point 2 places to the right. This changes 1.567 to 156.7.

Step 4: Write the Result

Therefore, the standard form of \(1.567 \times 10^2\) is 156.7.

Key Concepts

Standard Form

Standard form is a way of writing numbers that makes it easier to deal with very large or very small numbers. It is also known as scientific notation. This method involves expressing a number as the product of a number between 1 and 10, and a power of 10.

• The number part should always be greater than or equal to 1 and less than 10.
• The power of 10 indicates how many places the decimal point is moved.

For example, the number \(1.567 \times 10^2\) is in standard form. Here, 1.567 is between 1 and 10, and it is multiplied by \(10^2\), which translates to 100. This means when converting to a regular number, you shift the decimal point: \(1.567\) becomes 156.7 after moving the decimal point two places to the right.

Powers of Ten

In standard form, powers of ten help indicate how many places to move the decimal point. They are an exponent found on the number 10. When you see a positive exponent, you shift the decimal point to the right. This means you are multiplying by powers of ten, making the number larger.

• A positive power such as \(10^2\) means two spaces right, equivalent to multiplying by 100.
• A negative power, like \(10^{-2}\), would move the decimal to the left.

For example, with \(10^2\) in \(1.567 \times 10^2\), the result is 156.7 as you move the decimal point to the right two spaces (because of the 2 in \(10^2\)). Breaking down calculations using powers of ten is crucial to mastering standard form.

Decimal Notation

Decimal notation is the standard way of writing numbers using the base ten. It's what most people think of when they write numbers without any exponents. In our example, after converting \(1.567 \times 10^2\) using standard form rules, we wrote it as 156.7. This final expression, 156.7, is the decimal notation of the number.

• Decimal notation expresses numbers directly, without needing an exponent or multiplication.
• When converting from standard form, the end result should always be in decimal notation.

Understanding how to switch between decimal notation and standard form is essential. It allows you to work comfortably with numbers in various contexts.