Problem 28 Use the order of operations to s... [FREE SOLUTION]

Chapter 4: Problem 28

Use the order of operations to simplify each expression. $$\left(\frac{1}{6}+\frac{1}{3}\right)^{3}+\left(\frac{2}{5} \cdot \frac{3}{4}\right)^{2}$$

Short Answer

Expert verified

The expression simplifies to $\frac{43}{200}$.

Step by step solution

Simplify Inside Parentheses

We have two expressions inside parentheses to evaluate separately. For the first part, calculate the sum of fractions: \[\frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}\] For the second part, calculate the product of fractions: \[\frac{2}{5} \cdot \frac{3}{4} = \frac{6}{20} = \frac{3}{10}\]

Apply Exponents

Evaluate the exponents for the simplified expressions. The first expression raised to the power of 3: \[\left(\frac{1}{2}\right)^{3} = \frac{1}{8}\] The second expression raised to the power of 2: \[\left(\frac{3}{10}\right)^{2} = \frac{9}{100}\]

Add the Results

Add the results from Step 2: \[\frac{1}{8} + \frac{9}{100}\]To add these fractions, find a common denominator. The least common multiple of 8 and 100 is 400. Convert the fractions: \[\frac{1}{8} = \frac{50}{400}, \quad \frac{9}{100} = \frac{36}{400}\] Add the converted fractions: \[\frac{50}{400} + \frac{36}{400} = \frac{86}{400}\] Simplify the fraction: \[\frac{86}{400} = \frac{43}{200}\]

Simplification Check

Check that the fraction $\frac{43}{200}$ is in the simplest form by ensuring that the greatest common divisor (GCD) of 43 and 200 is 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fractions

Fractions represent a part of a whole and are written with two numbers: the numerator on top and the denominator below. When working with fractions in expressions, it's important to follow certain rules.

Adding fractions: To add fractions, their denominators must be the same. If the denominators are different, you need to find a common denominator before adding them. For example, $\frac{1}{6} + \frac{1}{3}$ requires converting $\frac{1}{3}$ to $\frac{2}{6}$ before adding to get $\frac{3}{6} = \frac{1}{2}$.
Multiplying fractions: This is more straightforward鈥攜ou simply multiply the numerators together and the denominators together. For instance, with $\frac{2}{5} \cdot \frac{3}{4}$, multiply 2 by 3 and 5 by 4 to get $\frac{6}{20}$, which simplifies to $\frac{3}{10}$.

Understanding these basics can help you tackle more complex fraction problems.

Exponents

Exponents indicate how many times a number, called the base, is multiplied by itself. They are expressed as a smaller number written to the top right of the base. In the expression given, exponents are applied to the results from the parentheses.

Calculating powers: For instance, to evaluate $\left(\frac{1}{2}\right)^3$, you multiply $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}$, which results in $\frac{1}{8}$.
Squared values: Squaring a number means multiplying it by itself, such as $\left(\frac{3}{10}\right)^2 = \frac{3}{10} \times \frac{3}{10} = \frac{9}{100}$.

Understanding how exponents work is crucial in simplifying mathematical expressions.

Common Denominators

To add or subtract fractions, they must have the same denominator. The denominator represents the total number of equal parts in a whole. When they don't match, you need to find a common denominator that is often the least common multiple (LCM) of the denominators.

LCM process: For $\frac{1}{8}$ and $\frac{9}{100}$, find the LCM of 8 and 100. The smallest number both can divide into is 400. So, convert these fractions: $\frac{1}{8} = \frac{50}{400}$ and $\frac{9}{100} = \frac{36}{400}$.
Adding with a common denominator: Once converted, you simply add the numerators: $\frac{50}{400} + \frac{36}{400} = \frac{86}{400}$.

Recognizing and using common denominators simplifies the addition and subtraction of fractions greatly.

Simplification

Simplification means reducing a fraction to its lowest terms so that the numerator and denominator are as small as possible. This often involves finding the greatest common divisor (GCD).

Finding the GCD: For the fraction $\frac{86}{400}$, you can simplify by finding the GCD of 86 and 400. Since they don't share any factors other than 1, the simplest form would be $\frac{43}{200}$.
Why simplify: Simplified fractions are easier to understand and work with. It's the cleanest way to present a final answer.

Simplification is a key step in ensuring that your final answers are concise and correct.

91影视

Use the order of operations to simplify each expression. $$\left(\frac{1}{6}+\frac{1}{3}\right)^{3}+\left(\frac{2}{5} \cdot \frac{3}{4}\right)^{2}$$

Short Answer

Step by step solution

Simplify Inside Parentheses

Apply Exponents

Add the Results

Simplification Check

Key Concepts

Fractions

Exponents

Common Denominators

Simplification

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