Question

For Exercises \(1-16\), match the power of 10 to its name or use. A. \(10^{-12}\) B. \(10^{-9}\) C. \(10^{-6}\) D. \(10^{-3}\) E. \(10^{3}\) F. \(10^{6}\) G. \(10^{9}\) H. \(10^{12}\) I. \(10^{15}\) Roughly equivalent to \(2^{10}\)

Short answer

The power of 10 that matches \(2^{10}\) is \(10^3\).

Step-by-step solution

Understand the Question

We need to match one of the given powers of 10 to the expression that is roughly equivalent to \(2^{10}\).

Calculate \(2^{10}\)

Calculate \(2^{10}\) which is equal to 1024. This can be obtained by repeatedly multiplying 2 by itself until you reach the 10th power: \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 1024\).

Compare with the Powers of 10

Now, compare 1024 with the given powers of 10: A. \(10^{-12}\), B. \(10^{-9}\), C. \(10^{-6}\), D. \(10^{-3}\), E. \(10^{3}\), F. \(10^{6}\), G. \(10^{9}\), H. \(10^{12}\), I. \(10^{15}\). 1024 is closest to \(10^3\) (which is 1000).

Conclusion

Since 1024 is very close to 1000, we can conclude that \(2^{10}\) is approximately equal to \(10^3\). Therefore, the answer corresponds to option E, \(10^{3}\).

Key concepts

Understanding Powers of 10

In mathematics, powers of 10 are used to express numbers as multiplied by 10 raised to a specific exponent. This method is incredibly useful when dealing with very large or very small numbers because it simplifies the notation and calculation processes. When we refer to a power of 10, we mean 10 multiplied by itself a certain number of times — this number is represented as the exponent. For example, when we write \(10^3\), it means \(10 \times 10 \times 10\), which equals 1000.
Similarly, negative exponents represent fractions or very small numbers. For example, \(10^{-3}\) equals \(\frac{1}{10^3}\) or \(\frac{1}{1000}\). Knowing how to interpret and use powers of 10 can help you to handle numbers in various scales easily.
Some commonly used powers of 10 include:

• \(10^3 = 1000\)
• \(10^6 = 1,000,000\)
• \(10^9 = 1,000,000,000\)

These powers form a foundation for understanding more complex mathematical concepts, including scientific notation.

Engaging with Math Exercises

Math exercises often aim to help students practice their understanding of concepts through application. In the context of powers of 10, exercises might include matching expressions, calculating exponents, or converting data into exponential form.
Such tasks require comprehension of how multiplication and division interact with powers as well as an ability to estimate and approximate. For instance, in the original exercise, the task was to match \(2^{10}\) with a power of 10. The key to solving it was noticing that \(2^{10} = 1024\), and 1024 is very close to \(10^3 = 1000\).
Through practice, math exercises enhance analytical and computational skills, encouraging learners to observe patterns and make informed estimations. They are fundamental in strengthening problem-solving abilities and applying theoretical knowledge in practical scenarios.

Navigating Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It leverages powers of 10 to simplify how we write and interpret these numbers.
A number in scientific notation is typically written as a product of a number between 1 and 10 (the coefficient) and a power of 10. For example:

• The number 1500 can be written in scientific notation as \(1.5 \times 10^3\).
• A very small number, like 0.00056, can be expressed as \(5.6 \times 10^{-4}\).

In scientific disciplines such as physics and chemistry, this notation is indispensable due to the frequent necessity of working with extremely large or tiny numbers.
Mastering scientific notation not only simplifies the arithmetic of very large or small numbers but also promotes ease in interpreting and conveying data across different fields of study.