Problem 50 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$(0.75)^{2}+\left(\frac{1}{4}\right)^{2}(7)$$

Step-by-Step Solution

Answer

The simplified expression equals 1.0.

Step 1: Square 0.75

First, we need to square the number 0.75. We calculate: \((0.75)^2 = 0.75 \times 0.75 = 0.5625\).

Step 2: Evaluate the Expression with Fractions

Next, handle the fractional part of the expression. Calculate \(\left(\frac{1}{4}\right)^2\): \(\left(\frac{1}{4}\right)^2 = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\).

Step 3: Multiply by 7

Use the result of Step 2 and multiply by 7 as specified in the expression: \(\frac{1}{16} \times 7 = \frac{7}{16}\).

Step 4: Convert Fraction to Decimal

Convert \(\frac{7}{16}\) to decimal form. Now, \(\frac{7}{16} = 0.4375\).

Step 5: Add the Decimals

Finally, we add the results from Step 1 and Step 4: \(0.5625 + 0.4375 = 1.0\).

Key Concepts

Square of a Number

Squaring a number simply means multiplying the number by itself. This is a common operation in arithmetic and algebra, useful in various calculations such as area measurement or when dealing with quadratic equations.
In the context of decimals, squaring involves careful multiplication to ensure precision. For example, squaring 0.75 involves multiplying 0.75 by itself:

• Performing this operation step-by-step:
  • First, multiply 0.75 by 0.75.
  • Calculating: \( 0.75 \times 0.75 = 0.5625 \).

It's vital to maintain decimal places to get an accurate result. This is how we get from the small steps to the complete squared value.Also, remember: when you square a decimal, the decimal point and its position affect the result significantly.

Fraction Multiplication

Multiplying fractions involves taking two fractions and finding a single fraction that represents their product. This requires multiplying the numerators (top numbers) with each other and the denominators (bottom numbers) with each other.Consider the example of squaring a fraction like \( \left( \frac{1}{4} \right)^2 \). Here's how it's done:

• Multiply the numerators:
  • 1 times 1 equals 1.
• Multiply the denominators:
  • 4 times 4 equals 16.

So, \( \left( \frac{1}{4} \right)^2 = \frac{1}{16} \).From there, if the expression asks for further multiplication by a whole number, as seen with \( \frac{1}{16} \times 7 \), follow the same rule:

• Multiply numerator by the whole number: 1 times 7 equals 7, keeping the denominator the same.

This process is simple but requires careful handling of numerators and denominators to keep them properly aligned.

Decimal Conversion

Decimal conversion is the process of changing a fraction into a decimal number. It's useful to improve readability and make further calculations easier. For example, to convert \( \frac{7}{16} \) into a decimal:

• You can perform division: 7 divided by 16.
• Using long division method: see how many times 16 fits into 7, factoring in decimals as needed.
• Calculate: \( 7 \div 16 = 0.4375 \).

Using decimals can simplify work because many people find arithmetic with decimals more intuitive than using fractions.In this specific exercise, decimals make the final operations clearer and speed up the addition at the end of our computations.