Problem 40 - Algebra and Trigonometry

Question

\(39-48=\) Simplify the expression. $$ \sqrt{75}+\sqrt{48} $$

Step-by-Step Solution

Answer

The simplified expression is \( 9\sqrt{3} \).

Step 1: Understanding the Expression

Our task is to simplify the expression \( \sqrt{75} + \sqrt{48} \). To do this, we need to simplify each square root separately by expressing them in terms of their prime factors.

Step 2: Simplifying \( \sqrt{75} \)

The number 75 can be expressed as a product of prime factors: \( 75 = 3 \times 5^2 \). The square root of \( 75 \) can be rewritten using these factors, resulting in \( \sqrt{75} = \sqrt{3 \times 5^2} \). This simplifies to \( 5\sqrt{3} \), because \( \sqrt{5^2} = 5 \).

Step 3: Simplifying \( \sqrt{48} \)

The number 48 can be expressed in prime factors as \( 48 = 2^4 \times 3 \). We can rewrite this under the square root: \( \sqrt{48} = \sqrt{2^4 \times 3} \). This simplifies to \( 4\sqrt{3} \), because \( \sqrt{2^4} = 4 \).

Step 4: Combining Terms

Now we have \( 5\sqrt{3} + 4\sqrt{3} \). Since both terms have the same radical part \( \sqrt{3} \), we can combine them just like adding like terms: \( 5\sqrt{3} + 4\sqrt{3} = 9\sqrt{3} \).

Step 5: Final Result

The simplified expression for \( \sqrt{75} + \sqrt{48} \) is \( 9\sqrt{3} \).

Key Concepts

Prime Factorization

Prime factorization is a method of breaking down a number into the most basic building blocks of its composition, which are prime numbers. A prime number is defined as any number greater than 1 that has no positive divisors other than 1 and itself. When we perform prime factorization on a number, we express it as a product of primes.
For example, to factorize 75, we divide it by the smallest prime number, which is 3:

• 75 divided by 3 gives 25.
• Next, 25 is divided by 5, resulting in 5.
• The process stops when you are left with a prime number that cannot be further divided by any other prime than itself: 25 is simply 5 times 5. So, 75 equals 3 times 5 squared.

Prime factorization helps us simplify square roots because it makes it easy to spot pairs of the same number. This is important for removing squares from inside the square root.

Like Terms

In algebra, like terms refer to terms that contain the same variable raised to the same power, including similar radical parts. This means you can combine them through addition or subtraction. Recognizing like terms is crucial when simplifying expressions.

• In the expression \(5\sqrt{3} + 4\sqrt{3}\), both terms have the \(\sqrt{3}\) part, making them like terms.
• Like terms can be combined by simply adding or subtracting their coefficients; in this case, 5 and 4.
• The result is \(9\sqrt{3}\) because \(5 + 4 = 9\).

Understanding the concept of like terms simplifies expressions and makes it easier to find solutions across many areas of algebra.

Square Root Properties

Square roots work on the principle of finding a number which, when multiplied by itself, results in the original number under the radical sign. There are important properties to remember when working with square roots:

• The square root of a product can be expressed as the product of the square roots, such as \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This helps in breaking down complex square roots into simpler components.
• If possible, factors appear in pairs and can be brought outside of the square root to simplify the expression. For example, \(\sqrt{5^2}\) becomes 5, because 5 times 5 equals 25, and the square root of 25 is 5.
• Similarly, \(\sqrt{2^4}\) turns into 4, because it equals \(\sqrt{16}\), and the square root of 16 is 4.

These properties make it easier to work with square roots in algebra by reducing more complex radical expressions to simpler ones that involve integers.