Problem 107 - Elementary Algebra

Question

\(-\frac{2}{7}+\frac{2}{7} \div\left(-\frac{1}{3}\right)^{2}\)

Step-by-Step Solution

Answer

\(\frac{16}{7}\)

Step 1: Identify the expression inside the parentheses

Locate the expression inside the parentheses. The expression to focus on is \(-\frac{1}{3}\).

Step 2: Square the expression inside the parentheses

Square \(-\frac{1}{3}\) to get: \[ \left( -\frac{1}{3} \right)^2 = \frac{1}{9} \]

Step 3: Divide the fraction by the squared value

Divide \(\frac{2}{7}\) by \(\frac{1}{9}\) to get: \[ \frac{2}{7} \div \frac{1}{9} = \frac{2}{7} \times \frac{9}{1} = \frac{18}{7} \]

Step 4: Simplify the entire expression

Evaluate \(-\frac{2}{7} + \frac{18}{7} \) by combining the fractions: \[ -\frac{2}{7} + \frac{18}{7} = \frac{-2 + 18}{7} = \frac{16}{7} \]

Key Concepts

Dividing Fractions

Dividing fractions might seem tricky, but it's straightforward once you know the steps. When you divide by a fraction, you are essentially multiplying by its reciprocal.
Start by identifying the fractions involved. For instance, if you want to divide \(\frac{2}{7}\) by \(\frac{1}{9}\), you first find the reciprocal of \(\frac{1}{9}\), which is \(\frac{9}{1}\).
Next, change the division sign to multiplication and multiply the fractions. So, \(\frac{2}{7} \bigg/ \frac{1}{9}\) becomes \(\frac{2}{7} \times \frac{9}{1} = \frac{18}{7}\).
Multiplying fractions is simply multiplying the numerators and denominators directly.
Remember these steps:

• Find the reciprocal of the fraction you are dividing by.
• Change the division sign to multiplication.
• Multiply the numerators and denominators.

This simplifies dividing fractions, making your calculations accurate and easy!

Squaring Fractions

Squaring a fraction involves multiplying the fraction by itself.
To square a fraction, such as \(-\frac{1}{3}\):

• Multiply the numerator by itself: \(-1 \times -1 = 1\)
• Multiply the denominator by itself: \(3 \times 3 = 9 \)

So, \(-\frac{1}{3} \bigg( -\frac{1}{3} \bigg) = \frac{1}{9}\).
It’s important to remember that squaring a negative fraction results in a positive fraction, as the sign is multiplied twice.
Practicing this will help in mastering other operations involving fractions.

Combining Fractions

Combining fractions, also known as adding or subtracting them, requires a common denominator.
For example, to combine \(-\frac{2}{7}\) and \(\frac{18}{7}\):

• Ensure the denominators are the same. Here, both fractions already have the same denominator, 7.
• Add (or subtract) the numerators directly. So, \(-2 + 18\) results in \(\frac{16}{7}\).

If the denominators are different, find a common denominator first before combining them.
This systematic approach ensures that adding or subtracting fractions is straightforward and accurate.
Practice makes combining fractions intuitive and helps in solving complex problems effortlessly.