Question

\begin{aligned} & 1.7 \times 10^{-3}--1.7 \times 10^3 \\ & \begin{array}{|l|l|} \hline \text { Select... } & \mathbf{7} \\ \hline> \\ \hline< \\ \hline= \\ \hline \end{array} \\ & \end{aligned}

Short answer

<

Step-by-step solution

Convert to Standard Form

Convert the scientific notation numbers to standard form to make the comparison easier. For the first number: \[1.7 \times 10^{-3} = 0.0017\]For the second number: \[1.7 \times 10^{3} = 1700\]

Compare the Numbers

Compare the two numbers in standard form.We have:\[0.0017 < 1700\]

Select the Correct Operator

Based on the comparison from Step 2, select the correct operator from the given options. Since \(0.0017\) is less than \(1700\), the correct operator is \(<\).

Key concepts

Standard Form

Understanding scientific notation is crucial for comparing large and small numbers easily. Scientific notation simplifies numbers by expressing them as a product of a coefficient and a power of 10. To convert a number from scientific notation to standard form, multiply the coefficient by 10 raised to the given power. For instance, the number \(1.7 \times 10^{-3}\) converts to standard form by moving the decimal point three places to the left, resulting in 0.0017. Similarly, \(1.7 \times 10^{3}\) means moving the decimal point three places to the right, resulting in 1700. Once in standard form, these numbers are easier to compare and work with in calculations.

Number Comparison

Comparing numbers in scientific notation becomes straightforward when they are converted to standard form. In standard form, you directly compare the values. For example, converting \(1.7 \times 10^{-3}\) gives 0.0017, and \(1.7 \times 10^{3}\) gives 1700. When you look at these two numbers, it is clear that 0.0017 is much smaller than 1700. Using this method allows you to easily determine which number is larger or smaller without dealing with the complexities of scientific notation. Always ensure both numbers are in standard form before making a comparison.

Mathematical Operators

Mathematical operators like \(>\), \(<\), and \(=\) are used to show the relationship between two numbers. Once numbers are in standard form, selecting the correct operator is easy. For example, with numbers 0.0017 and 1700, the correct operator is \(<\). Hence, we write 0.0017 \(<\) 1700. Understanding and using the correct operator helps in representing mathematical relationships accurately. Here's a quick guide:

• Use \(>\) if the first number is greater than the second.
• Use \(<\) if the first number is smaller than the second.
• Use \(=\) if both numbers are equal.

By practicing these comparisons, you become more comfortable with understanding the size and scale of numbers in various forms.