Question

The distance between two planets is \(4,560,000,000\) miles. Express this in scientific notation. (A) \(4.56 \times 10^6\) (B) \(4.56 \times 10^7\) (C) \(456 \times 10^7\) (D) \(4.56 \times 10^9\)

Short answer

The correct answer is (D) \(4.56 \times 10^9\).

Step-by-step solution

Identify the number in scientific notation form

Express \(4,560,000,000\) in the form \(a \times 10^b\), where \(1 \leq a < 10\) and \(b\) is an integer.

Determine the value of the decimal 'a'

To find the value of 'a', move the decimal point from the end of the number (4,560,000,000) to the first (non-zero) digit. This will give us: \(4.56\).

Count the number of decimal places we moved

Count the places we moved the decimal point: In going from \(4,560,000,000\) to \(4.56\), we moved the decimal point 9 places to the left.

Determine the value of the exponent 'b'

Since we moved to the left, the exponent 'b' will be positive. The value of 'b' will be equal to the number of decimal places we moved: \(b = 9\)

Rewrite the number in scientific notation

Using the information from Steps 2 and 4, we can rewrite the number as: \(4.56 \times 10^9\).

Identify the correct answer from the given options

We found that the given number in scientific notation is \(4.56 \times 10^9\). This corresponds to option (D). Thus, the correct answer is (D) \(4.56 \times 10^9\).

Key concepts

Math in Everyday Life

Mathematics education is a cornerstone of our education system, offering invaluable skills for daily life. Understanding concepts like scientific notation not only aids in tackling large numbers but is also practical in fields such as science and engineering.

Learning mathematics strengthens problem-solving skills. It encourages logical thinking and the ability to analyze complex situations. Through homework exercises like converting numbers to scientific notation, students enhance their ability to work through puzzles efficiently.

It's crucial to apply math concepts to real-life situations. This not only motivates students but also makes learning experiences relatable and engaging.

• Applying math in physics to calculate planetary distances.
• Using math in chemistry to measure reactant quantities.
• Integrating math in economics to understand financial data.

In sum, mastering math concepts prepares students for both academic success and practical life challenges.

Understanding Exponents

Exponents are powerful tools used in mathematics to express numbers more efficiently. They tell us how many times to multiply a number by itself.

For instance, in the expression \(10^9\), it indicates that 10 is multiplied by itself 9 times, resulting in 1,000,000,000. This is a much simpler way to write such a large figure.

Understanding how exponents work is vital for converting numbers into scientific notation. When dealing with huge or tiny numbers, exponents simplify writing and comprehension.

• Positive exponent: indicates multiplication (e.g., \(10^3 = 1000\)).
• Negative exponent: indicates division (e.g., \(10^{-3} = 0.001\)).
• Zero exponent: any non-zero base raised to zero is 1 (e.g., \(10^0 = 1\)).

Grasping these concepts makes understanding and manipulating numbers straightforward and efficient.

Number Representation Techniques

Number representation involves expressing numbers in a form that is easier to understand and use. Scientific notation is one special form to achieve this.

In scientific notation, numbers are expressed as a product of their significant digits and a power of ten, making both very large and very tiny numbers manageable. For example, the number 4,560,000,000 becomes \(4.56 \times 10^9\).

This method is particularly useful in scientific fields, where precision is crucial, and figures are often extreme in size. By representing numbers this way, we simplify calculations, comparisons, and comprehension.

• Facilitates working with astronomical or microscopic measurements.
• Increases clarity in data communication.
• Makes calculations more straightforward and minimizes errors.

Ultimately, understanding how to represent numbers efficiently is integral in mathematics and science, promoting clarity and ease of use.