Adding Square Roots $$ \sqrt{50} + \sqrt{8} $$

Operations with Radicals: Adding Square Roots

In this lesson, we focus on operations with radicals, specifically adding square roots. Before radicals can be added, each square root must be simplified as much as possible. Only like radicals can be combined.

1. Theory of Adding Square Roots

Square roots can only be added together if they have the same radical part. This means the expression inside the square root must be identical. To achieve this, we often need to simplify each radical first.

2. Concept Explanation

Simplifying a square root involves factoring the number inside the radical into a perfect square multiplied by another number. The square root of the perfect square becomes a whole number, while the remaining factor stays inside the radical.

3. Formula or Steps

1. Simplify each square root separately.
2. Check that the radicals are the same.
3. Add the coefficients.
4. Write the final simplified result.

4. Example Problem

Problem:

$$ \sqrt{50} + \sqrt{8} $$

Solution:

Step 1: Simplify each square root.

$$ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} $$ $$ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} $$

Step 2: Add the like radical terms.

$$ 5\sqrt{2} + 2\sqrt{2} = (5 + 2)\sqrt{2} $$ $$ = 7\sqrt{2} $$

Final Answer:

$$ \sqrt{50} + \sqrt{8} = 7\sqrt{2} $$

Operations with Fractions $$ \frac{43}{30} = A + \frac{1}{B + \frac{1}{C + \frac{1}{D}}} $$

Operations with Fractions: Continued Fractions

In this lesson, we explore a more advanced operation with fractions by working with continued fractions. The goal is not only to simplify a fraction, but also to identify specific values hidden within its structure.

1. Problem Overview

We are given the following expression:

$$ \frac{43}{30} = A + \frac{1}{B + \frac{1}{C + \frac{1}{D}}} $$

Our task is to determine the values of A, B, C, and D, then find the value of A + B + C + D.

2. Step-by-Step Solution

Step 1: Convert the fraction into a mixed number.

$$ \frac{43}{30} = 1 + \frac{13}{30} $$

This works because a mixed number can be written as the sum of a whole number and a fraction.

Step 2: Convert the fraction into a continued fraction.

Recall the rule:

$$ \frac{a}{b} = \frac{1}{\frac{b}{a}} $$

Applying this rule to the fraction:

$$ \frac{13}{30} = \frac{1}{\frac{30}{13}} $$

Now simplify the denominator step by step:

$$ \frac{30}{13} = 2 + \frac{4}{13} $$ $$ \frac{4}{13} = \frac{1}{\frac{13}{4}} = \frac{1}{3 + \frac{1}{4}} $$

So the continued fraction becomes:

$$ \frac{13}{30} = \frac{1}{2 + \frac{1}{3 + \frac{1}{4}}} $$

3. Identifying the Values

Substituting back into the original expression:

$$ \frac{43}{30} = 1 + \frac{1}{2 + \frac{1}{3 + \frac{1}{4}}} $$

From this form, we can identify:

• A = 1
• B = 2
• C = 3
• D = 4

4. Final Calculation

Now add all the values:

$$ A + B + C + D = 1 + 2 + 3 + 4 = 10 $$

Final Answer:

$$ A + B + C + D = 10 $$

Dividing Fractions $$ 5 \div \frac{3}{4} $$

Dividing Fractions with Whole Numbers

Dividing fractions is an important skill in mathematics and follows a clear set of rules. Unlike addition or subtraction, division requires changing the operation before solving. Understanding this process makes fraction division much easier and more meaningful.

1. Theory of Dividing Fractions

When dividing by a fraction, we do not divide directly. Instead, we change the division into multiplication by using the reciprocal of the fraction. The reciprocal is found by swapping the numerator and the denominator.

2. Concept Explanation

The division problem “five divided by three fourths” can be interpreted as asking: “How many groups of three fourths are in five?” This way of thinking helps us understand why multiplication by the reciprocal works.

By counting how many times three fourths fits into five whole units, we can clearly see the meaning behind the final answer.

3. Formula or Steps

1. Change the division into multiplication by the reciprocal.
2. Multiply the numerators and denominators.
3. Simplify the result.
4. Convert to a mixed number if the fraction is improper.

4. Example Problem

Problem:

$$ 5 \div \frac{3}{4} $$

Solution:

Step 1: Change division into multiplication by the reciprocal.

$$ 5 \div \frac{3}{4} = 5 \times \frac{4}{3} $$

Step 2: Multiply the numerators and denominators.

$$ 5 \times \frac{4}{3} = \frac{5 \times 4}{1 \times 3} = \frac{20}{3} $$

Step 3: Simplify the result.

$$ \frac{20}{3} = 6 \frac{2}{3} $$

5. Conceptual Visualization

Conceptually, dividing five by three fourths asks how many groups of three fourths can be formed from five. We can fit six complete groups of three fourths, with two thirds of another group remaining. This confirms the final result.

Final Answer:

$$ 5 \div \frac{3}{4} = \frac{20}{3} = 6 \frac{2}{3} $$