Problem 132 - Elementary and Intermediate Algebra

Question

Translate the set of instructions to an expression and then evaluate it. Increase the square of the reciprocal of \(-2\) by the difference of \(-0.25\) and \(-1\)

Step-by-Step Solution

Answer

\(\frac{1}{2}\)

Step 1: Square of the Reciprocal

First, find the reciprocal of -2 which is -\frac{1}{2}. Then, square it:\(- \left(\frac{1}{2}\right)^2 = -\frac{1}{4}\).

Step 2: Calculate the Difference

Find the difference between \(-0.25\) and \(-1\):\(-0.25 - (-1) = -0.25 + 1 = 0.75\).

Step 3: Increase by the Difference

Add the result from Step 2 to the result of Step 1:\(-\frac{1}{4} + 0.75\).Convert \(0.75\) to fractions:\(0.75 = \frac{3}{4}\).Add the fractions:\(-\frac{1}{4} + \frac{3}{4} = \frac{2}{4} = \frac{1}{2}\).

Key Concepts

Reciprocal

Understanding the reciprocal of a number is essential when dealing with algebraic expressions. A reciprocal is simply the flipped version of the number—if we start with a fraction like \( a/b \), its reciprocal is \( b/a \). For whole numbers, the reciprocal is 1 divided by the number. For instance, the reciprocal of 2 is \( 1/2 \).

In our exercise, we found the reciprocal of \(-2\), which is \(-1/2\). It's important to carefully handle the negative sign, as it stays with the reciprocal, just like it does with the original number. Consider these points:

• Reciprocal of a positive integer is positive.
• Reciprocal of a negative integer is negative.
• Reciprocal of zero does not exist because division by zero is undefined.

Once you have the reciprocal, you can proceed with other operations like squaring.

Squaring Numbers

Squaring numbers means multiplying the number by itself. It's a fundamental concept that pops up frequently in algebra. For example, squaring 3 simply means \(3 \times 3\), which equals 9.

In our problem, we squared the reciprocal of \(-2\), which is \(-1/2\). This becomes \((-1/2) \times (-1/2)\). When multiplying two negative numbers, the result is positive, so the answer is \(1/4\).

Some key points when squaring numbers:

• Squaring a positive number results in a positive number.
• Squaring a negative number also results in a positive number.
• Squaring zero results in zero.

Understanding squaring helps in various complex calculations, such as those involving roots and powers.

Fractions Arithmetic

Handling fractions correctly is crucial in mathematics, especially when performing operations like addition, subtraction, multiplication, or division.

When you're adding or subtracting fractions, it’s necessary to have a common denominator. This ensures that we are adding or subtracting parts of the same size.

In our problem, we converted \(0.75\) to a fraction, \(\frac{3}{4}\), to allow it to be easily added to \(-\frac{1}{4}\). Here are some key considerations:

• To add or subtract fractions, they must have the same denominator.
• If denominators differ, convert fractions to have a common denominator.
• Multiply numerators and denominators separately when raising fractions to a power or multiplying them.

This approach promotes accuracy and makes solving problems much easier. In our case, subtracting \(-\frac{1}{4}\) from \(\frac{3}{4}\) gave us \(\frac{1}{2}\). Such operations, though simple, are foundational in understanding and manipulating algebraic expressions.