Problem 22 - Prealgebra

Question

Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$8 \sqrt{48 x^{3}}+2 \sqrt{12 x^{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \(36x \sqrt{3x}\).

Step 1: Understanding the Expression

We are given the expression \(8 \sqrt{48 x^3} + 2 \sqrt{12 x^3}\). Our task is to simplify the square roots and combine them if possible. We need to break down the numbers inside the square roots and simplify each part separately.

Step 2: Simplifying \(\sqrt{48 x^3}\)

First, factor 48 into its prime factors: \(48 = 16 \cdot 3 = 4^2 \cdot 3\). Rewrite \(\sqrt{48x^3}\) as \(\sqrt{16 \cdot 3 \cdot x^3}\). Extract the square root of 16 and part of \(x^3\): \(\sqrt{16} = 4\), and \(\sqrt{x^3} = x^{3/2} = x\sqrt{x}\). Thus, \(\sqrt{48 x^3} = 4x \sqrt{3x}\).

Step 3: Simplifying \(\sqrt{12 x^3}\)

Next, factor 12 into its prime factors: \(12 = 4 \cdot 3\). Rewrite \(\sqrt{12x^3}\) as \(\sqrt{4 \cdot 3 \cdot x^3}\). Extract the square root of 4 and part of \(x^3\): \(\sqrt{4} = 2\), and \(\sqrt{x^3} = x^{3/2} = x\sqrt{x}\). Thus, \(\sqrt{12 x^3} = 2x \sqrt{3x}\).

Step 4: Substituting Simplified Parts

Substitute the simplified forms back into the original expression: \(8 \times 4x \sqrt{3x} + 2 \times 2x \sqrt{3x}\). This becomes \(32x \sqrt{3x} + 4x \sqrt{3x}\).

Step 5: Combining Like Terms

Both terms have \(x \sqrt{3x}\) in common, so we can combine them: \((32x + 4x)\sqrt{3x} = 36x \sqrt{3x}\).

Step 6: Final Answer

The simplified expression is \(36x \sqrt{3x}\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into a product of prime numbers. Prime numbers are numbers that can only be evenly divided by 1 and themselves, such as 2, 3, 5, 7, etc.
To perform prime factorization, start by dividing the number by the smallest prime number possible, and continue the process with the quotient until you end up with all prime numbers.

• For example, let’s factorize 48. Start with the smallest prime, which is 2:
  • 48 divided by 2 equals 24
  • 24 divided by 2 equals 12
  • 12 divided by 2 equals 6
  • 6 divided by 2 equals 3 (3 is a prime number)
  This gives us the prime factorization: \[48 = 2^4 \times 3\].
• For 12, the factorization would be:
  • 12 divided by 2 equals 6
  • 6 divided by 2 equals 3 (3 is a prime number)
  Hence, the prime factorization is: \[12 = 2^2 \times 3\].

Prime factorization is a key step when simplifying square roots, as it helps in identifying perfect squares that can be extracted for simplification. This is evident in our exercise where simplifying \(\sqrt{48x^3}\) and \(\sqrt{12x^3}\) involves recognizing their prime factors.

Combining Like Terms

In mathematics, combining like terms refers to the process of adding or subtracting terms that have identical variable parts. This is crucial because it allows you to simplify expressions for a clearer result. The terms must have the exact same variables and exponents to be considered "like terms."

• In our example, after simplifying the square roots, we end up with terms like \(32x \sqrt{3x}\) and \(4x \sqrt{3x}\).
• These terms are "like" because they both have the same variable part, \(x \sqrt{3x}\).
• To combine these terms, add the numerical coefficients (in this case, 32 and 4) while keeping the variable part unchanged.

So, \(32x \sqrt{3x} + 4x \sqrt{3x} = (32+4)x \sqrt{3x} = 36x \sqrt{3x}\).
Combining like terms simplifies the expression to its most concise form and is particularly useful when solving algebraic equations or simplifying expressions, which is what we aimed to achieve in the given exercise.

Extracting Square Roots

Extracting square roots is the process of finding the number or expression that, when multiplied by itself, gives the original number or expression under the square root. It's about identifying perfect squares within the number so they can be "extracted" or "taken out" of the square root.

• Consider the term \(\sqrt{48x^3}\). First, break it down into prime factors: \(48 = 16 \cdot 3\) and \(16 = 4^2\). This makes \(\sqrt{48x^3} = \sqrt{16 \cdot 3 \cdot x^3}\).
• The \(\sqrt{16}\) simplifies to 4 since 16 is a perfect square. The \(x^3\) is simplified as \(x^{3/2} = x\cdot\sqrt{x}\).
• This gives us \(4x \sqrt{3x}\) for \(\sqrt{48x^3}\).

In our exercise, this process was essential for simplifying both \(\sqrt{48x^3}\) and \(\sqrt{12x^3}\) in order to extract their respective square roots. It allows simplification by reducing the square root to its simplest radical form. This means recognizing and removing perfect squares so the expression beneath the square root sign is as small as possible.

• With \(\sqrt{12x^3}\), the same approach applies: \(12 = 4 \cdot 3\).
• Extract the square root of \(4\) as \(2\) and simplify the \(x^3\) similarly to get \(2x \sqrt{3x}\).

Extracting square roots makes expressions easier to manage and allows us to combine like terms effectively as demonstrated in simplifying the exercise.