Question

Complete these problems in scientific notation. Round to the correct number of significant figures. a. \(\left(5.31 \times 10^{-2} \mathrm{cm}\right) \times\left(2.46 \times 10^{5} \mathrm{cm}\right)\) b. \(\left(3.78 \times 10^{3} \mathrm{m}\right) \times\left(7.21 \times 10^{2} \mathrm{m}\right)\) c. \(\left(8.12 \times 10^{-3} \mathrm{m}\right) \times\left(1.14 \times 10^{-5} \mathrm{m}\right)\) d. \(\left(9.33 \times 10^{4} \mathrm{mm}\right) \div\left(3.0 \times 10^{2} \mathrm{mm}\right)\) e. \(\left(4.42 \times 10^{-3} \mathrm{kg}\right) \div\left(2.0 \times 10^{2} \mathrm{kg}\right)\) \(\mathrm{f}\left(6.42 \times 10^{-2} \mathrm{g}\right) \div\left(3.21 \times 10^{-3} \mathrm{g}\right)\)

Short answer

The short answers to the given problems are as follows: a. \(1.31 \times 10^3 \mathrm{cm}\) b. \(2.73 \times 10^5 \mathrm{m}\) c. \(9.26 \times 10^{-8} \mathrm{m}\) d. \(3.11 \times 10^2 \mathrm{mm}\) e. \(2.21 \times 10^{-5} \mathrm{kg}\) f. \(2.0 \times 10^1 \mathrm{g}\)

Step-by-step solution

Multiply the significant digits

First, multiply the significant digits: \(5.31 \times 2.46 = 13.062\)

Add the exponents of 10

Add the exponents of 10: \(-2 + 5 = 3\)

Write the result

The result is \(1.31 \times 10^3 \mathrm{cm}\), rounded to the correct number of significant figures. #b. Calculate the product of two numbers in scientific notation:

Multiply the significant digits

First, multiply the significant digits: \(3.78 \times 7.21 = 27.2498\)

Add the exponents of 10

Add the exponents of 10: \(3+2=5\)

Write the result

The result is \(2.73 \times 10^5 \mathrm{m}\), rounded to the correct number of significant figures. #c. Calculate the product of two numbers in scientific notation:

Multiply the significant digits

First, multiply the significant digits: \(8.12 \times 1.14 = 9.2568\)

Add the exponents of 10

Add the exponents of 10: \(-3 + (-5) = -8\)

Write the result

The result is \(9.26 \times 10^{-8} \mathrm{m}\), rounded to the correct number of significant figures. #d. Calculate the quotient of two numbers in scientific notation:

Divide the significant digits

First, divide the significant digits: \(9.33 ÷ 3.0 = 3.11\)

Subtract the exponents of 10

Subtract the exponents of 10: \(4 - 2 = 2\)

Write the result

The result is \(3.11 \times 10^2 \mathrm{mm}\), rounded to the correct number of significant figures. #e. Calculate the quotient of two numbers in scientific notation:

Divide the significant digits

First, divide the significant digits: \(4.42 ÷ 2.0 = 2.21\)

Subtract the exponents of 10

Subtract the exponents of 10: \(-3 - 2 = -5\)

Write the result

The result is \(2.21 \times 10^{-5} \mathrm{kg}\), rounded to the correct number of significant figures. #f. Calculate the quotient of two numbers in scientific notation:

Divide the significant digits

First, divide the significant digits: \(6.42 ÷ 3.21 = 2\)

Subtract the exponents of 10

Subtract the exponents of 10: \(-2 - (-3) = 1\)

Write the result

The result is \(2.0 \times 10^1 \mathrm{g}\), rounded to the correct number of significant figures.

Key concepts

Significant Figures

Significant figures are the digits in a number that contribute to its precision. These figures help indicate the accuracy of a measurement or calculation. When working with scientific notation, it's important to identify the number of significant figures, which includes

• All non-zero digits (e.g., 5, 1, 9),
• Any zeros between significant figures (e.g., 305 or 7.03),
• Any trailing zeros in a decimal number (e.g., 3.00 or 45.700).

By rounding to the correct number of significant figures, you ensure that your scientific notation is precise. For example, in the calculations a. the number 5.31 has three significant figures. So, when you multiply it by another number, your final answer must maintain this level of precision.

Exponents

Exponents in scientific notation simplify the representation of very large or very small numbers by using powers of 10. For instance, in the number 3.78 × 10⁵, the exponent 5 shows that the base 10 is multiplied five times. This allows the representation of 378,000 in a compact form. Similarly, negative exponents indicate division, such as 5.31 × 10⁻², which means 5.31 is divided by 100. Understanding how to work with exponents is crucial as they indicate how much you should move the decimal point, based on whether the exponent is positive or negative.

Multiplication and Division of Scientific Notation

To multiply numbers in scientific notation

• Multiply the significant digits first.
• Add the exponents.

For instance, in (a) you first multiply 5.31 and 2.46 to get 13.062. Then, add the exponents: -2 + 5 = 3 resulting in the expression 1.31 × 10³ by rounding to significant figures. For division,

• Divide the significant digits, and
• Subtract the exponents.

For example, if you have (d) 9.33 ÷ 3.0, get 3.11 for significant figures, and subtract exponents: 4 - 2 = 2, yielding 3.11 × 10².

Rounding in Scientific Notation

Rounding is essential for keeping the answers precise in scientific notation. The rule generally follows the usual rounding process:

• If the next number is 5 or more, round up.
• If it's less than 5, round down.

For example, the result of a is 13.062, but for three significant figures, this rounds to 1.31 × 10³. This step ensures that the precision matches the least precise factor in the multiplication or division calculations. Consistently rounding helps maintain a clear level of accuracy in scientific results.

Calculations with Scientific Notation

Calculating with scientific notation requires systematically handling both the significant digits and the powers of ten. When given expression e, follow these steps:

• Divide the coefficients, here 4.42 ÷ 2.0 = 2.21, then,
• Determine the exponent by subtracting the exponents, which results in -3 - 2 = -5.

This yields a final result of 2.21 × 10⁻⁵ kg. Such a methodical approach allows you to perform complex calculations with ease and precision, ensuring both the base and the exponent are correctly handled.