Problem 212 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{5}{6} \sqrt{162}+\frac{3}{16} \sqrt{128} $$

Step-by-Step Solution

Answer

9 \sqrt{2}

Step 1: Simplify \( \sqrt{162} \)

First, find the prime factorization of 162.\( 162 = 2 \times 81 = 2 \times (9 \times 9) = 2 \times (3 \times 3) \times (3 \times 3) = 2 \times 3^4 \)Thus, \(\sqrt{162} = \sqrt{2 \times 3^4} = \sqrt{2 \times (3^2)^2} = 3^2 \sqrt{2} = 9 \sqrt{2} \)

Step 2: Simplify \(\sqrt{128} \)

Now, find the prime factorization of 128.\(128 = 2^7 \)Thus,\(\sqrt{128} = \sqrt{2^7} = \sqrt{(2^6) \times 2} = 2^3 \sqrt{2} = 8 \sqrt{2} \)

Step 3: Substitute and combine like terms

Substitute \( \sqrt{162} \) and \( \sqrt{128} \) back into the expression:\( \frac{5}{6} \sqrt{162} + \frac{3}{16} \sqrt{128} = \frac{5}{6} \times 9 \sqrt{2} + \frac{3}{16} \times 8 \sqrt{2} \)Simplify coefficients:\(\frac{5}{6} \times 9 = \frac{45}{6} = 7.5 \)and\(\frac{3}{16} \times 8 = \frac{24}{16} = 1.5 \)

Step 4: Combine the simplified terms

Add the simplified terms:\(7.5 \sqrt{2} + 1.5 \sqrt{2} = (7.5 + 1.5) \sqrt{2} = 9 \sqrt{2} \)

Key Concepts

Prime Factorization

Prime factorization is breaking down a number into its basic building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that are divisible only by 1 and themselves.
For example:

• The prime factorization of 162 is found by dividing by the smallest prime number possible, repeatedly:$$162 = 2 \times 81 = 2 \times (3 \times 27) = 2 \times 3^4$$
• The prime factorization of 128 involves dividing by 2 repeatedly until reaching a prime number:$$128 = 2^7$$

Understanding prime factorization is crucial for simplifying radical expressions, as it helps in breaking down the numbers under the radicals.

Square Roots

Square roots help in finding numbers that, when multiplied by themselves, give the original number. For instance, the square root of 9 is 3 because $$3 \times 3 = 9$$.
For simplifying radical expressions, it’s useful to express the number inside the square root as a product of squares where possible:

• \rightarrow For $$\frac{5}{6} \times \text{√162}$$, first we express 162 in terms of its prime factors:
  $$\text{√162} = \text{√(2 \times 3^4)}$$
  \text{With further simplification, we get} $$3^2 \times \text{√2} = 9\text{√2}$$
  This step converts the original square root expression into a product of an integer and a simpler square root.
• \rightarrow Similarly, for $$\frac{3}{16}\times\text{√128}$$, we represent 128 as:
  $$\text{√128} = \text{√(2^7)} = 2^3 \times \text{√2} = 8\text{√2}$$
  This simplifies it into a product of an integer and a simpler square root.

Algebraic Simplification

Algebraic simplification techniques help to make expressions less complex but equivalent in value. Here’s how we simplify our example effectively:

• \rightarrow Substitute the simplified radicals back into the expression:
  $$\frac{5}{6} \times 9 \text{√2} + \frac{3}{16} \times 8 \text{√2}$$
• \rightarrow Simplify the coefficients separately:
  $$\frac{5}{6} \times 9 = 7.5$$ and$$ \frac{3}{16} \times 8 = 1.5$$

This step helps to turn the initially complicated fractions into simple multiplications. Using these techniques removes complexity from expressions, making them easier to understand and work with.

Combining Like Terms

Combining like terms is a basic algebraic principle for simplifying expressions. Terms that have the same variable part are considered like terms and can be combined:

• For example: Combine $$7.5 \text{√2}$$ and $$1.5 \text{√2}$$ by adding their coefficients:
  $$7.5 + 1.5 = 9$$

This results in: $$9 \text{√2}$$

This principle is critical for simplifying overall expressions by reducing them to a single, more manageable term.