Problem 100 - Chemistry and Chemical Reactivity

Question

Evaluate the value of \(x\) in the following expressions: (a) \(x=\left[\left(9.345 \times 10^{-4}\right)\left(6.23 \times 10^{6}\right)\right]^{3}\) (b) \(x=\sqrt{\left(1.23 \times 10^{-2}\right)\left(4.5 \times 10^{5}\right)}\) (c) \(x=\sqrt[3]{\left(1.23 \times 10^{-2}\right)\left(4.5 \times 10^{5}\right)}\) Show the answers to the correct number of significant figures.

Step-by-Step Solution

Answer

(a) 1.973 × 10^{11}, (b) 23.4, (c) 17.7.

Step 1: Evaluate Expression (a)

First, calculate the product inside the brackets:\[(9.345 \times 10^{-4})(6.23 \times 10^{6}) = 9.345 \times 6.23 \times 10^{-4+6} = 58.20835 \times 10^{2} = 5.820835 \times 10^{3}\]Next, raise this result to the power of 3:\[(5.820835 \times 10^{3})^{3} = (5.820835)^{3} \times 10^{9} = 197.293 \times 10^{9}\]Simplify to the correct number of significant figures (4 significant figures):\[1.973 \times 10^{11}\]

Step 2: Evaluate Expression (b)

Calculate the product inside the square root:\[(1.23 \times 10^{-2})(4.5 \times 10^{5}) = 1.23 \times 4.5 \times 10^{-2+5} = 5.535 \times 10^{3}\]Then calculate the square root:\[\sqrt{5.535 \times 10^{3}} \approx \sqrt{5.535} \times 10^{1.5} = 2.3526 \times 10^{1.5}\]Round to 3 significant figures:\[23.4\]

Step 3: Evaluate Expression (c)

Calculate the product inside the cube root, which is the same as in (b):\[5.535 \times 10^{3}\]Now calculate the cube root:\[\sqrt[3]{5.535 \times 10^{3}} \approx \sqrt[3]{5.535} \times 10^{1} \approx 1.7714 \times 10^{1}\]Round to 3 significant figures:\[17.7\]

Step 4: Concluding the Solution

After determining the values for each part of the problem, confirm and present the final answers rounded to the appropriate number of significant figures as discussed.

Key Concepts

Scientific Notation

Scientific notation is a clever way of writing very large or very small numbers. It mostly involves expressing a number as a product of a number between 1 and 10 and a power of ten. This method allows for easy handling and communication of numbers that are otherwise lengthy or cumbersome.

Here is how it works:

• Let's say you have a number like 9,345,000. In scientific notation, it becomes \(9.345 \times 10^{6}\).
• Similarly, a small number such as 0.0009345 can be written as \(9.345 \times 10^{-4}\).

Scientific notation helps to make calculations, like multiplication and division, more manageable because it separates the numerical factor from the order of magnitude factor (the power of ten).
You simply multiply the decimal parts and handle the exponents using the laws of exponents. For instance, when multiplying \((9.345 \times 10^{-4})\) and \((6.23 \times 10^{6})\), you multiply the coefficients (\(9.345 \times 6.23\)) and add the exponents (-4 + 6) for the power of 10.

Cube Root

A cube root is the opposite of cubing a number (i.e., raising a number to the power of three). Finding the cube root of a number gives you a value that, when multiplied by itself twice, will return the original number. If the equation is \(x^3 = a\), then \(x\) is the cube root of \(a\).

Key facts about cube roots:

• The cube root of \(8\) is \(2\) because \(2^3 = 8\).
• Cubes and cube roots are useful in calculating volumes and solving cubic equations.

In scientific problems, you may encounter cube roots involving numbers in scientific notation. For instance, the expression \(\sqrt[3]{5.535 \times 10^{3}}\) involves finding the cube root of both the numeric coefficient 5.535 and the power of ten \(10^{3}\).

Break it down as follows:

• Cube root of 5.535 is approximately 1.7714.
• Cube root of \(10^{3}\) is \(10\) because \(\sqrt[3]{10^{3}} = 10\).

Multiplying these results gives you the cube root value of \(17.7\).

Square Root

Taking the square root of a number involves finding a value that, when squared, results in the original number. It is one of the most fundamental operations in mathematics and has vast applications.

Here's an understanding of square roots:

• For example, the square root of 25 is 5, since \(5^2 = 25\).
• Square roots are used in geometry, particularly when working with areas.

When numbers in scientific notation are involved, the approach is to find the square root of both the coefficient and the power of ten separately. Consider the equation \(\sqrt{5.535 \times 10^{3}}\).

• Square root of 5.535 is approximately 2.3526.
• Square root of \(10^{3}\) is \(10^{1.5}\) (since we take half of the exponent \(3\) to get \(1.5\)).

After calculating these, multiply the outcomes to determine the square root, which results in an approximate value, and rounded off, it would be 23.4 to three significant figures.