Problem 46 - Prealgebra

Question

Simplify each expression. $$\left(\frac{13}{21}-\frac{13}{35}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{676}{11025} \).

Step 1: Subtract Fractions

To simplify the expression \( \left(\frac{13}{21} - \frac{13}{35}\right)^2 \), start by finding the common denominator to subtract the two fractions. The denominators are 21 and 35. The least common multiple is 105.Convert each fraction:\[ \frac{13}{21} = \frac{13 \times 5}{105} = \frac{65}{105} \]\[ \frac{13}{35} = \frac{13 \times 3}{105} = \frac{39}{105} \]Now subtract these fractions:\[ \frac{65}{105} - \frac{39}{105} = \frac{65 - 39}{105} = \frac{26}{105} \]

Step 2: Square the Result

After finding the difference, we have to square \( \frac{26}{105} \).Square the fraction:\[ \left(\frac{26}{105}\right)^2 = \frac{26^2}{105^2} = \frac{676}{11025} \]

Step 3: Simplify the Fraction

Finally, check if \( \frac{676}{11025} \) can be simplified. Both numbers are already the smallest form for the given result, so the expression is fully simplified.Therefore, the simplified form of the original expression is:\[ \frac{676}{11025} \]

Key Concepts

Common Denominator

To subtract fractions, having the same denominator is essential. This shared denominator is known as a "common denominator." When denominators differ, you must find a number that is a multiple of both, referred to as the "least common multiple" (LCM). For the fractions \( \frac{13}{21} \) and \( \frac{13}{35} \), the LCM of 21 and 35 is 105.
This means we convert each fraction to an equivalent one with 105 as the denominator:

• For \( \frac{13}{21} \), multiply both numerator and denominator by 5, resulting in \( \frac{65}{105} \).
• For \( \frac{13}{35} \), multiply both by 3, giving \( \frac{39}{105} \).

By converting each fraction to have 105 as the denominator, they become easier to subtract directly. This simplifies the operation greatly, allowing straightforward subtraction.

Subtracting Fractions

Once fractions have a common denominator, subtraction becomes simpler. Look at the fractions \( \frac{65}{105} \) and \( \frac{39}{105} \). They now have the same denominator, so subtract their numerators:

• Subtract 39 from 65, which equals 26.

Therefore, \( \frac{65}{105} - \frac{39}{105} = \frac{26}{105} \).
The denominator, 105, stays the same after subtraction. Subtracting fractions only affects the numerators when the denominators are already common. Keep the denominators intact and apply the operation only to numerators for simplicity and accuracy.

Squaring Fractions

Squaring a fraction involves multiplying it by itself. After subtraction, you have \( \frac{26}{105} \). To square it, follow these steps:

• Square the numerator: \( 26 \times 26 = 676 \).
• Square the denominator: \( 105 \times 105 = 11025 \).

Hence, \( \left(\frac{26}{105}\right)^2 = \frac{676}{11025} \).
Squaring fractions results in a fraction whose numerator and denominator have been squared. This generates another fraction representing the square of the original. Thus, squaring expands the fraction's size, often requiring simplification to its most basic form.

Simplifying Fractions

Simplifying fractions involves reducing them to their smallest possible form. This means dividing the numerator and the denominator by their greatest common divisor (GCD).
In the case of \( \frac{676}{11025} \), both numbers are already in their simplest form because they share no common divisor other than 1. Sometimes, it requires trial or using specific algorithms like the Euclidean to determine the GCD.

• Check divisibility by smaller prime numbers.
• Use the factorization method for large numbers if necessary.

Once confirmed simplified, the fraction \( \frac{676}{11025} \) is the final, simplified result of squaring the initial expression.