Problem 22 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{90 x^{2} y^{3}}$$

Step-by-Step Solution

Answer

\(3xy\sqrt{10y}\)

Step 1: Prime Factorization of 90

First, we need to break down 90 into its prime factors. This will help us simplify the square root. The prime factorization of 90 is done as follows: start by dividing by 2, the smallest prime number. \[ 90 \div 2 = 45 \] Next, divide 45 by 3: \[ 45 \div 3 = 15 \] Continue with 15, dividing by 3 again: \[ 15 \div 3 = 5 \] Finally, 5 is a prime number. So, the complete prime factorization of 90 is \[ 90 = 2 \times 3^2 \times 5 \]

Step 2: Rewrite the Expression Under the Radical

Using the prime factorization, rewrite the expression under the square root. \[ \sqrt{90 x^{2} y^{3}} = \sqrt{2 \cdot 3^2 \cdot 5 \cdot x^2 \cdot y^3} \]

Step 3: Simplify the Radical Expression

To simplify, we take out complete squares from under the radical. Since square roots take out pairs of identical factors, identify and take out these pairs.\[ \sqrt{2 \cdot 3^2 \cdot 5 \cdot x^2 \cdot y^3} = 3 \cdot x \cdot y^{1}\cdot \sqrt{2 \times 5 \times y} = 3xy\sqrt{10y}\] Here, \(3^2\) gives a \(3\), \(x^2\) gives an \(x\), and \(y^2\) (from \(y^3\)) gives a \(y\), leaving \(y\) under the square root together with factors \(2\) and \(5\), which combine into \(10\).

Step 4: Final Simplified Form

Based on the steps above, present the expression in its fully simplified form: \[ 3xy\sqrt{10y} \]

Key Concepts

Prime Factorization

To simplify square roots, one handy method is using prime factorization. This helps in breaking down a number into its prime components, which means numbers that are only divisible by 1 and themselves. Exploring prime factorization allows us to understand the number more deeply and make it easier to work with radicals.
Here is how you can do it:

• Start with the smallest prime number, which is 2. Check if your number is divisible by 2 and divide if possible.
• Move to the next prime number (3, 5, 7, etc.) to divide the result further until it turns into a prime number itself.

For example, to factorize 90, we would divide it through primes as follows:
- Divide by 2, to get 45 - Divide 45 by 3, to get 15 - Keep dividing 15 by 3, to end up with 5 - Since 5 is prime, our prime factors are 2, 3, and 5.
Expressing 90 using these is crucial for managing and simplifying radical expressions, which is what we explore next.

Radical Expression

A radical expression involves the square root notation that is often used to show the root of a number involving prime factors. Square roots are special because they "undo" squaring, reducing numbers into simpler components when possible.
More specifically, a radical expression like \(\sqrt{90 x^{2} y^{3}}\) makes use of prime factorization to rewrite the expression such that multiplication or division occur under the square root.
This process is as follows:

• Break down the entire expression using factorization into prime numbers and variables as seen in the example \(\sqrt{2 \cdot 3^2 \cdot 5 \cdot x^2 \cdot y^3}\).
• Manage the number and variables separately to make it easier for simplifications.

Recognizing when parts of the expression can be "paired" under this symbol is integral because from there, removal of these pairs becomes more feasible, which we will see in the next concept of simplifying radicals further.

Simplifying Radicals

Having broken down the expression, the next step is simplifying radicals by removing perfect squares from under the square root. This means you find sets of identical factors in the prime factorization and move them outside the radical.

• Look for pairs among the factors. For example, in \(3^2\), the pair allows moving "3" outside the radical.
• Apply this process to variables as well. If, for instance, you have \(x^2\), the "x" moves outside the square root.

In our case, with \(\sqrt{90 x^{2} y^{3}\): - \(3^2\) goes to \(3\), \(x^2\) turns to \(x\), and from \(y^3\), \(y^2\) gives an exterior \(y\), leaving behind unnecessary factors under the radical.So the expression reduces to \(3xy\sqrt{10y}\), simplifying your radical approach dramatically by focusing on pairs and leftovers for a cleaner outcome.