Problem 24 - Algebra 2

Question

Divide and simplify. Assume that all variables are positive. $$ \frac{\sqrt{48 x^{3}}}{\sqrt{3 x y^{2}}} $$

Step-by-Step Solution

Answer

The simplified form of \(\frac{\sqrt{48 x^{3}}}{\sqrt{3 x y^{2}}}\) is \(4x/y\).

Step 1: Apply the Division Rule for Radicals

The division rule tells us that we can combine the two radicals into one by performing the division under a single square-root sign. Therefore, \(\sqrt{48 x^{3}}/\sqrt{3 x y^{2}}\) becomes \(\sqrt{(48 x^{3})/(3 x y^{2})}\).

Step 2: Perform the Division Inside the Square Root

When dividing with variables, remember to divide the coefficients (number parts) and the variable parts separately. So, perform the division \( (48/3)\), the division \(x^{3}/x\), and also recognise that there is no similar term to cancel out \(y^{2}\). The expression becomes \(\sqrt{(16 x^{2}/y^{2})}\).

Step 3: Simplify the Square Root

Since we have a perfect square under the radical, we can simplify by taking the square root of the coefficient and of the variables. Remember that the square root of a square is just the original number or variable (assuming all variables are positive), so the expression simplifies to \(4x/y\).

Key Concepts

Division of Radicals

Dividing radicals can seem tricky, but with some simple rules, it becomes easy to understand. The division rule of radicals states that when you divide two square roots, you can combine them under a single square root sign. For example, if you have \(\frac{\sqrt{a}}{\sqrt{b}}\), this is the same as \(\sqrt{\frac{a}{b}}\).
This rule allows you to simplify the expression by combining the terms and makes it much easier to work with complex radical expressions. Always remember: the number or expression inside the square root should be simplified as much as possible for clarity.
When applying this to variables, ensure that each variable follows the same rule, so if they appear in both the numerator and denominator, try to simplify them.

Simplifying Radicals

Once you have done the division under the radical sign, the next step is to simplify the expression further. Simplifying radicals often means finding perfect squares within the numbers and variables under the square root.
For example, if you find a \( \sqrt{16}\), since 16 is a perfect square (\(4 \times 4\)), it simplifies to 4 on its own. Using this approach, look for numbers or expressions under the radical that can be square-rooted perfectly.

• If the number is a perfect square, take its square root.
• If the variables are raised to an even power, like \(x^2\), take the square root to remove the exponent.

This allows you to express the radical in a simpler, more manageable form.

Variable Division

Variable division within radicals brings another layer of simplification. When dividing like variables within an expression, apply basic rules of exponents. If you have \(x^a/x^b\), it simplifies to \(x^{a-b}\). It works just like numbers - you simply subtract the exponents.
This helps break down the expression into simpler parts, which makes taking further steps, like simplifying the radical, easier.
Consider the impact of any given variable throughout the expression. Important details such as ensuring that no division by zero occurs and that all resulting variables are positive will ensure the correctness and simplification of your resultant solution.

• Simplify using the basic laws of exponents.
• Make sure variables remaining in the expression are positive, which simplifies handling within radicals.

Understanding these fundamental principles can greatly make handling radicals, even those with variables, more intuitive and approachable.