Question

Express each value in exponential form. Where appropriate, include units in your answer. (a) solar radiation received by Earth: 173 thousand trillion watts (b) average human cell diameter: 1 ten-millionth of a meter (c) the distance between the centers of the atoms in silver metal: 142 trillionths of a meter (d) \(\frac{\left(5.07 \times 10^{4}\right) \times\left(1.8 \times 10^{-3}\right)^{2}}{0.065+\left(3.3 \times 10^{-2}\right)}=\)

Short answer

\n(a) Solar radiation: \(1.73 \times 10^{26}\) watts \n(b) Average human cell diameter: \(1.0 \times 10^{-7}\) meters \n(c) Distance between atoms in Silver: \(1.42 \times 10^{-10}\) meters \n(d) Computed Value = \(7.28 \times 10^{10}\)

Step-by-step solution

Express Solar Radiation

Given Solar radiation received by Earth: 173 thousand trillion watts. We will express this value in exponential format giving the answer as \(1.73 \times 10^{26}\) watts

Express Average Human Cell Diameter

Given Average human cell diameter: 1 ten-millionth of a meter. We express this value in exponential format giving the result as \(1.0 \times 10^{-7}\) meters

Express Distance between Atoms in Silver

Given Distance between the centers of atoms in silver metal: 142 trillionths of a meter. Converting this into exponential notation, we get \(1.42 \times 10^{-10}\) meters

Evaluate the Given Expression

We will simplify the given expression \(\frac{\left(5.07 \times 10^{4}\right) \times\left(1.8 \times 10^{-3}\right)^{2}}{0.065+\left(3.3 \times 10^{-2}\right)}\), by performing operations in the order recommended by the order of operations, BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, Subtraction). After performing the operations in the correct order, we get \(7.28 \times 10^{10}\)

Key concepts

Solar Radiation

Solar radiation is the energy emitted by the sun in the form of electromagnetic waves. It is crucial for sustaining life on Earth as it provides heat and light. A common question often involves understanding how much solar radiation reaches Earth. This value is typically very large, measured in watts, which is a unit of power.

- In our case, the solar radiation received by Earth is given as 173 thousand trillion watts. This is equivalent to 173,000,000,000,000,000, or in scientific terms, we simplify it to exponential form as \(1.73 \times 10^{26}\) watts.

Converting such vast numbers into exponential notation is essential to make them more manageable and easier to visualize. This helps scientists and students communicate and process such large quantities without dealing with numerous zeros.

Human Cell Diameter

The human cell is a fundamental unit of life and measuring it accurately is essential for biological and medical research. Cells have various shapes and sizes, but we can express the average diameter of a human cell using exponential notation for clarity.

- For example, the average human cell diameter might be described as one ten-millionth of a meter. In numerical terms, this is 0.0000001 meters. In exponential notation, it is expressed as \(1.0 \times 10^{-7}\) meters.

This format not only simplifies the writing but also helps keep calculations precise and prevents potential errors that might arise from handling small scales in biological studies.

Atomic Distance in Silver

Silver is a metal composed of closely packed atoms. The distance between the centers of these atoms is crucial in determining the properties of silver. Understanding atomic distances helps us explore material characteristics and how they interact at an atomic level.

- In the case of silver, we find that this distance is 142 trillionths of a meter. This is numerically presented as 0.000000000142 meters. Converting into exponential notation, it becomes \(1.42 \times 10^{-10}\) meters.

The use of exponential notation for atomic distances is quite helpful in physics and chemistry, as it provides a clean, readable format that simplifies complex measurements.

Order of Operations

The order of operations is a fundamental rule in mathematics to solve expressions consistently and accurately. Often remembered by the acronym "BEDMAS": Brackets, Exponents, Division, Multiplication, Addition, and Subtraction.

- For example, let's consider the expression \(\frac{(5.07 \times 10^{4}) \times (1.8 \times 10^{-3})^{2}}{0.065+(3.3 \times 10^{-2})}\). To solve this correctly, we must follow the order of operations:

• First, address the brackets and solve any operations inside them.
• Then, calculate any exponents.
• Next, perform division and multiplication as they appear from left to right.
• Finally, execute addition and subtraction in sequence from left to right.

By following these steps, the expression simplifies to \(7.28 \times 10^{10}\). Observing this order is critical to arriving at the correct solution to mathematical problems.