Problem 36 - Contemporary Precalculus

Question

Simplify the expression without using a calculator. $$\sqrt{150}+\sqrt{24}$$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\sqrt{150}+\sqrt{24}$$ without using a calculator. Answer: $$7\sqrt{6}$$

Step 1: Prime Factorization of 150 and 24

To find the prime factors of 150 and 24, we can use the method of repeated division by prime numbers. The prime factors of 150 are 2, 3, and 5. The prime factors of 24 are 2 and 3. So we can write: $$150 = 2\times3\times5^2 \quad and \quad 24 = 2^3\times3$$

Step 2: Simplify the square roots

Next, replace the numbers inside the square root with their prime factors. $$\sqrt{150}+\sqrt{24} = \sqrt{2\times3\times5^2}+\sqrt{2^3\times3}$$ Now, find the square roots of perfect squares inside the square roots: $$\sqrt{150}+\sqrt{24} = \sqrt{2\times3\times(5^2)}+\sqrt{(2^2) \times 2\times3} $$ $$\sqrt{150}+\sqrt{24} = 5\sqrt{2\times3}+2\sqrt{2\times3} $$

Step 3: Combine like terms

Now we can combine the terms with the same square roots inside: $$\sqrt{150}+\sqrt{24} = (5+2)\sqrt{2\times3}$$ Which simplifies to: $$\sqrt{150}+\sqrt{24} = 7\sqrt{6}$$ The simplified expression without using a calculator is $$7\sqrt{6}$$.

Key Concepts

Prime Factorization

Prime factorization is a method used to break down a number into a product of its prime numbers. A prime number is a natural number greater than 1, and it can only be divided by 1 and itself without leaving a remainder. In this exercise, we apply prime factorization to simplify expressions under square roots.

To find the prime factors of 150, divide by the smallest prime number, 2, and continue with 3, and then 5 until you only have prime numbers. For 150, we get: 150 = 2 x 3 x 5 x 5. This can also be written as 150 = 2 x 3 x 5².

• Start with 150: divide by 2 (150 ÷ 2 = 75).
• Continue with 75: divide by 3 (75 ÷ 3 = 25).
• Finally, divide 25 by 5 which primes out 5².

For 24, you follow a similar process: 24 = 2 x 2 x 2 x 3, or 24 = 2³ x 3. This systematic breakdown helps isolate and identify the prime components of any given number, streamlining further mathematical operations like taking square roots.

Square Roots

Square roots simplify expressions by pulling out perfect squares. Perfect squares are numbers that are entire squares of an integer, like 4, 9, and 16. Once numbers are broken down into prime factors, it becomes easier to spot these perfect squares inside the radical.

Using the prime factorization results, let's simplify the square roots in the expression \(\sqrt{150} + \sqrt{24}\). We already know:

• \(150 = 2 \times 3 \times 5^2\), meaning \(\sqrt{150} = \sqrt{2 \times 3 \times 5^2}\).
• \(24 = 2^3 \times 3\), meaning \(\sqrt{24} = \sqrt{2^3 \times 3}\).

From here, extract the perfect squares:

• In \(\sqrt{150}\), \(5^2\) is a perfect square and becomes 5 when taken out of the square root.
• In \(\sqrt{24}\), \(2^2\) is a perfect square and becomes 2 when taken out.

Thus, \(\sqrt{150} = 5\sqrt{6}\) and \(\sqrt{24} = 2\sqrt{6}\). This step is crucial for simplifying radical expressions and plays a significant role in combining terms later.

Combining Like Terms

After breaking down and simplifying the square roots, the next step in simplifying radical expressions is to combine like terms. Like terms are terms within an expression that have the same variable or radicals.

In our expression \(\sqrt{150} + \sqrt{24}\), we previously simplified to \(5\sqrt{6}\) and \(2\sqrt{6}\). Both terms share the same radical, \(\sqrt{6}\). To combine them, you simply add or subtract their coefficients, much like combining coefficients in algebra.

• Add their coefficients: \(5\sqrt{6} + 2\sqrt{6} = (5+2)\sqrt{6}\).
• This results in \(7\sqrt{6}\).

Combining like terms helps in making the expression more manageable and easier to understand. It's essentially grouping and simplifying, which is an important part of solving algebraic expressions involving radicals. This process beautifully showcases the power of simplification in mathematics.