Problem 76 - Chemistry and Chemical Reactivity

Question

Carry out the following operations. Provide the answer with the correct number of significant figures. (a) \(\left(6.25 \times 10^{2}\right)^{3}\) (b) \(\sqrt{2.35 \times 10^{-3}}\) (c) \(\left(2.35 \times 10^{-3}\right)^{1 / 3}\)

Step-by-Step Solution

Answer

(a) \(2.44 \times 10^{8}\), (b) \(4.85 \times 10^{-2}\), (c) \(0.134\).

Step 1: Understanding Significant Figures

Before we solve the problems, let's understand that the number of significant figures is determined by the original measurement, not the result. Generally, any operations (multiplication, division, exponentiation) will carry the lowest number of significant figures from the operands involved. This will guide the precision of our final answers.

Step 1: Calculating and Rounding (a)

The operation requires us to compute \((6.25 \times 10^{2})^{3}\). First, calculate \((6.25)^{3} = 244.140625\). The number of significant figures in 6.25 is three, so our answer must have three significant figures. Therefore, 244.140625 is rounded to 244. Finally, calculate the power of ten: \((10^{2})^{3} = 10^{6}\). Thus, the final answer is \(2.44 \times 10^{8}\).

Step 2: Calculating and Rounding (b)

For \(\sqrt{2.35 \times 10^{-3}}\), first calculate \(\sqrt{2.35} = 1.532232\). The number 2.35 has three significant figures, so our answer must be given in three significant figures: 1.53. Then calculate \(\sqrt{10^{-3}} = 10^{-1.5}\). Thus, the complete answer is \(1.53 \times 10^{-1.5}\) or approximately \(4.85 \times 10^{-2}\) when expressed in scientific notation.

Step 3: Calculating and Rounding (c)

The operation \((2.35 \times 10^{-3})^{1 / 3}\) requires calculating \((2.35)^{1/3} = 1.336504\). Keep three significant figures: 1.34. Then calculate \((10^{-3})^{1/3} = 10^{-1}\). So the final answer is \(1.34 \times 10^{-1}\), which can also be written \(0.134\) when expressed in standard decimal notation.

Key Concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too big or small to conveniently write in decimal form. It's particularly useful in science due to the vast ranges of quantities involved. The general format is:

• A number between 1 and 10
• Multiplied by an integer power of 10

For instance, in the expression \(6.25 \times 10^{2}\), "6.25" is the coefficient, while "10" is the base raised to the power of "2," which is the exponent. Here are few points to remember:

• When multiplying, you add the exponents.
• When dividing, you subtract the exponents.
• Exponents can be positive or negative; a negative exponent indicates a small fraction.

This notation helps in simplifying calculations, especially when dealing with large numbers, like in our original exercise problem.

Exponents and Roots

Exponents are numbers that say how many times to use a number in a multiplication, while roots do the opposite operation. For example, \((6.25)^{3}\) means 6.25 is multiplied by itself two more times. Similarly, \(\sqrt{2.35} = (2.35)^{1/2}\) involves finding a number that when multiplied by itself gives 2.35.

• A power like \(10^{2}\) is a simple form of exponent, indicating 10 multiplied by itself once.
• When raising a number to a fraction, like \((2.35)^{1/3}\), you are finding a cube root.
• The laws of exponents, such as \(a^{m} \times a^{n} = a^{m+n}\), are valuable for rearranging and simplifying calculations.

Understanding these concepts is crucial for solving equations involving powers and roots efficiently.

Rounding Rules

Rounding is the process of reducing the digits of a number while trying to keep its value similar. You might round up or down based on specific rules, and how it's applied can impact significant figures. In problems like our exercise, rounding ensures the result reflects the precision of the initial measurements.

• If the first non-significant digit is 5 or higher, round up.
• Otherwise, round down.
• Ensure the result in calculations obeys the rule concerning the least precise measurement.

For instance, when the number has three significant figures, like "6.25," the result also must be rounded to three significant figures. This maintains consistency and accuracy in scientific work.