Problem 49 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$(0.25)^{2}+\left(\frac{1}{4}\right)^{2}(3)$$

Step-by-Step Solution

Answer

The simplified result is 0.25.

Step 1: Square 0.25

Calculate the square of 0.25. \( (0.25)^2 = 0.25 \times 0.25 = 0.0625 \)

Step 2: Square \( \frac{1}{4} \)

Square the fraction \( \frac{1}{4} \).\( \left( \frac{1}{4} \right)^2 = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \)

Step 3: Multiply by 3

Multiply the result from Step 2 by 3.\( \left( \frac{1}{16} \right) \times 3 = \frac{3}{16} \)

Step 4: Convert to Decimal

Convert \( \frac{3}{16} \) to a decimal.\( 0.1875 \)

Step 5: Add the Results

Add the decimal results from Step 1 and Step 4.\( 0.0625 + 0.1875 = 0.25 \)

Key Concepts

Decimal Operations Explained

Decimal operations involve basic arithmetic functions applied to decimal numbers, such as addition, subtraction, multiplication, and division. When dealing with decimal operations, it's essential to line up the decimal points to ensure accuracy.
For addition and subtraction, line up the decimal points and ensure that you insert zeros where necessary to keep the columns even.

• Add or subtract the numbers as you would whole numbers.
• Put the decimal point in the answer directly under the other decimal points.

Multiplication of decimals requires you to multiply the numbers ignoring the decimals initially, and then count the total number of decimal places in both the factors to place the decimal accurately in the answer. The division of decimals, however, involves moving the decimal point to convert the divisor into a whole number and making the same adjustment to the dividend. Then, divide as you normally would and place the decimal directly above its position in the dividend. It's crucial to maintain the correct placement of decimal points throughout, ensuring precise results.

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. The base is the number being multiplied, and the exponent indicates how many times the base is used as a factor in the multiplication.
For example, in the expression \( (0.25)^2 \), 0.25 is the base, and 2 is the exponent. This tells us to multiply 0.25 by itself: \( 0.25 \times 0.25 = 0.0625 \).

When dealing with fractions, like \( \left( \frac{1}{4} \right)^2 \), follow the same principle. Multiply the fraction's numerator and denominator separately by themselves: \( \left( \frac{1}{4} \right) \times \left( \frac{1}{4} \right) = \frac{1}{16} \).
Exponents can apply to any mathematical term, including whole numbers, fractions, and decimals, ensuring you understand how to multiply each part accurately is key to solving exponential problems.

Simplifying Expressions

Simplifying expressions involves performing all possible operations within the expression to reduce it to its simplest form. It's like tidying up a mathematical statement. In simplification, you need to carefully perform all multiplications and divisions before moving onto addition and subtraction.
In the given example, you first handle exponents, followed by multiplication, and finally, addition of the resulting values. When working with fractions, converting them to decimals often makes the operations more straightforward. Converting \( \frac{3}{16} \) to a decimal (0.1875) made adding it to previous decimal results easier.
Always remember:

• Follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) - commonly remembered as PEMDAS.
• Consistently apply decimal precision throughout the process to maintain accuracy.
• Simplifying not only reduces the expression but makes it easier to interpret and use in further calculations.

This practice is an essential skill that helps cultivate a more profound understanding of mathematical operations and expressions.