Problem 32 - Algebra 2

Question

$$ \frac{15 \sqrt{60 x^{5}}}{3 \sqrt{12 x}} $$

Step-by-Step Solution

Answer

The simplified expression is \(5x^2\sqrt{5x}\).

Step 1: Break down numbers into their prime factors and simplify the fraction

First, we need to simplify the fraction by dividing both numerator and denominator by their greatest common divisor. Here, the greatest common divisor is 15, so we divide both by this number to get: \(\frac{5 \sqrt{60 x^{5}}}{\sqrt{12 x}}\)

Step 2: Simplify square roots

Next, we need to simplify the square roots. We can do that by breaking down numbers under the square root into their prime factors. For \(60 = 2^2 * 15\) and \(12 = 2^2 * 3\). When simplifying square roots, we group the common prime factors together and take the ones outside the square root that have a pair. We must consider also x while doing this. \(x^5 = x^4 * x\) and \(x^4 = (x^2)^2\). So, finally the expression becomes : \(5x^2\sqrt{15x/3}\)

Step 3: Simplification

Finally, simplify the expression under the root and the root itself. It becomes: \(5x^2\sqrt{5x}\).

Key Concepts

Prime Factorization

When we talk about breaking down numbers into their prime factors, we mean expressing a number as the product of its basic building blocks, known as prime numbers. A prime number is a number that has no other divisors apart from 1 and itself. For instance, the prime factorization of 60 means finding which primes multiply together to make 60.

Here's how we do it for 60:

• 60 divides evenly by 2 (since it's even). 60 divided by 2 equals 30.
• Continue with 30, which also divides by 2 to get 15.
• 15 is divisible by 3, resulting in 5, which is a prime number.

So, 60 can be expressed as: \[60 = 2^2 \times 3 \times 5\]The same process applies to any integer. Breaking numbers into their prime factors aids in simplifying complex expressions and equations effectively. This process is crucial, especially when working with square roots, which we'll simplify next.

Square Roots Simplification

Simplifying square roots involves expressing the root in its simplest possible terms. We achieve this by breaking down the number inside the square root into its prime factors and taking pairs out.

For example, consider the number 60, whose prime factorization is:\[60 = 2^2 \times 3 \times 5\]From here, we can extract pairs:

• The number 2 appears twice, so one 2 can come out of the root.
• 3 and 5 do not pair, so they stay inside.

Thus, \(\sqrt{60}\) simplifies to \(2 \sqrt{15}\).Similarly, for a variable like \(x^5\):

• Break it into \(x^4 \times x = (x^2)^2 \times x\).
• The \(x^2\) is squared, so one \(x^2\) comes out of the root.

This yields: \(x^2 \sqrt{x}\).Simplifying square roots using this method makes math problems easier to manage and solve!

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. Finding the GCD helps in simplifying fractions and expressions, like what we saw in the original problem.

In the problem, the numbers 15 (numerator) and 3 (denominator) share a GCD of 3. To identify this, look at each number's factors:

• The factors of 15 are: 1, 3, 5, and 15.
• The factors of 3 are: 1 and 3.

The largest factor common to both is 3. Thus, you divide both the numerator and denominator by 3 to simplify the fraction.

GCD finds critical use beyond just numbers – it's also helpful with variable expressions. Looking for common powers in terms allows us to simplify algebraic expressions effectively. Using this method ensures cleaner, simplified forms of mathematical expressions, preparing us better for advanced math challenges.