Problem 40 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ 5 \sqrt{2 x y^{6}} \cdot 2 \sqrt{2 x^{3} y} $$

Step-by-Step Solution

Answer

The simplified expression is \(20x^{2}y^{3}\sqrt{x y}\)

Step 1: Multiply the Coefficients and Radicands

First, we need to multiply the constants (coefficient) outside of the radicals and the radicands (terms inside the radicals) separately. This gives us \(10 \sqrt{4x^{4}y^{7}}\).

Step 2: Simplify the Radicand

Next, look for perfect squares within the radicand in order to simplify the expression. In this case, \(4x^{4}y^{6}\) can be taken out of the square root. Now we get \(10 \cdot 2x^{2}y^{3} \sqrt{x y}\) which simplifies to \(20x^{2}y^{3}\sqrt{x y}\).

Step 3: Rationalize the Denominator

In this problem, we aren't given a fraction, so there is no need to rationalize the denominator. Please skip this step.

Key Concepts

multiplying radicals

Multiplying radicals might sound like a complicated task at first, but it's simpler than you think. When you multiply radicals, you need to keep two things in mind: multiply the coefficients (the numbers outside the square root), and multiply the radicands (the numbers or expressions inside the square root). In our example, we have the expression \(5 \sqrt{2xy^6} \cdot 2 \sqrt{2x^3y}\). Here, the coefficients are 5 and 2. So you multiply them together: \(5 \times 2 = 10\).Next, take care of the radicands, \(2xy^6\) and \(2x^3y\). Multiply them inside the square root: \(2 \times 2 = 4\), \(x \times x^3 = x^4\), and \(y^6 \times y = y^7\). This gives us \(\sqrt{4x^4y^7}\). After these steps, the expression becomes \(10 \sqrt{4x^4y^7}\). Quite simple, isn't it? Just make sure to multiply both the coefficients and the radicands separately! Remembering this will help you with any similar problems you encounter.

simplifying radicands

Simplifying radicands is all about making the expression inside the square root as simple as possible. The goal is to pull out any perfect squares, which are numbers or expressions that are "the same" times another "the same", like \(4 = 2 \times 2\) or \(x^4 = (x^2)^2\).For \(\sqrt{4x^4y^7}\), look closely at each part:

• 4 is a perfect square since \(2^2 = 4\).
• \(x^4\) is a perfect square because \((x^2)^2 = x^4\).
• \(y^7\) is a bit tricky. You can take out \(y^6\) as \((y^3)^2\), leaving one \(y\) under the square root.

Taking these out of the square root, our expression becomes: \(2x^2y^3 \sqrt{xy}\). This step can greatly simplify complex expressions, so it's worth practicing to get comfortable with recognizing and extracting perfect squares from radicands.

expression simplification

Expression simplification involves reducing any algebraic expression to its most compact form without changing its meaning. This makes the expression cleaner and often easier to work with.In the given example, the simplified expression after multiplying and simplifying the radicands is \(10 \cdot 2x^2y^3 \sqrt{xy}\). To simplify further: multiply 10 and 2, getting \(20x^2y^3 \sqrt{xy}\).When you simplify expressions:

• Check if you can combine like terms.
• Ensure all the exponents are simplified.
• Write expression without unnecessary parentheses inside or outside the radicals.

This process helps you to manage longer expressions with ease and can simplify your calculations tremendously.

square roots

The square root symbol \(\sqrt{\cdot}\) denotes finding a value that, when multiplied by itself, gives the original number or expression. It’s essential to understand it, as many algebra problems include square roots. When simplifying square roots, you often see expressions where the contents of the root, like \(4x^4\), can be broken down to smaller parts or even "escaped" out of the root.For example, when you have \(\sqrt{4x^4}\), notice:

• The square root of 4 is 2, because \(2 \times 2 = 4\).
• For \(x^4\), the square root is \(x^2\), as \((x^2)^2 = x^4\).

Understanding square roots in this way allows you to break them down into manageable pieces, making complex expressions easier to simplify. Always look out for factors you know are perfect squares – they can be your shortcut to faster simplification!