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Chapter 5: Problem 256

Simplify. \(\left(\frac{1}{2}\right)^{2}+(2.1)(8.3)\)

Short Answer

Expert verified
17.68

Step by step solution

01

- Simplify the First Term

Evaluate \(\frac{1}{2}\)^{2}. First, square the numerator and the denominator: \(\frac{1^{2}}{2^{2}} = \frac{1}{4}\). So, \(\frac{1}{2}\)^{2} = \(\frac{1}{4}\).
02

- Simplify the Second Term

Multiply the decimal numbers \(2.1\) and \(8.3\). \(2.1 \times 8.3 = 17.43\).
03

- Combine the Results

Add the results of both terms: \(\frac{1}{4} + 17.43\).
04

- Convert Fraction to Decimal

Convert \(\frac{1}{4}\) to a decimal, which is \(0.25\).
05

- Perform the Final Addition

Add \(0.25\) to \(17.43\): \(0.25 + 17.43 = 17.68\).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is all about breaking down a mathematical expression into its simplest form.
To do this, follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition, and Subtraction).
In our exercise, the first expression is \(\frac{1}{2}\)^2, which means squaring the fraction. First, we square the numerator (1) and the denominator (2) to get \(\frac{1}{4}\).
This process helps in reducing the complexity of the expression step by step.
Fraction to Decimal Conversion
Often in mathematics, you'll need to convert a fraction to a decimal for easier calculation.
For example, \(\frac{1}{4}\) can be converted to 0.25 because:
- Divide the numerator (1) by the denominator (4) to get 0.25.
This step is crucial when you need to add or multiply fractions with decimals as it simplifies further operations.
Multiplying Decimals
Multiplying decimals involves straightforward steps. To multiply 2.1 by 8.3 in our exercise, ignore the decimal points:
- Multiply 21 by 83 to get 1743.
- Since each original number had one decimal place, place the decimal point in the result: 17.43.
It's important to count the decimal places correctly to ensure an accurate result.
Addition of Mixed Number Formats
Sometimes, mathematical problems involve adding fractions and decimals.
To make it simple, first convert the fraction to a decimal.
For instance, \(\frac{1}{4}\) becomes 0.25.
Then, add the decimal values together:
- Add 0.25 (from \(\frac{1}{4}\)) to 17.43 to get 17.68.
This process ensures that the addition is straightforward and accurate.

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