Problem 25 - Algebra 2

Question

Divide and simplify. Assume that all variables are positive. $$ \frac{\sqrt{56 x^{5} y^{5}}}{\sqrt{7 x y}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \( \frac{\sqrt{56 x^{5} y^{5}}}{\sqrt{7 x y}} \) is \( 2 \sqrt{2} x^{2} y^{2} \).

Step 1: Break down each square root

Rewrite \( \sqrt{56 x^{5} y^{5}} \) as \( \sqrt{56} \sqrt{x^{5}} \sqrt{y^{5}} \) and \( \sqrt{7 x y} \) as \( \sqrt{7} \sqrt{x} \sqrt{y} \). When you do this, the division problem becomes \( \frac{\sqrt{56} \sqrt{x^{5}} \sqrt{y^{5}}}{\sqrt{7} \sqrt{x} \sqrt{y}} \).

Step 2: Simplify each square root separately

Simplify \( \sqrt{56} \) to \( 2 \sqrt{14} \) (since \( 56 = 4 \times 14 \) and \( \sqrt{4} = 2 \)). \( \sqrt{x^{5}} \) simplifies to \( x^{2} \sqrt{x} \), and \( \sqrt{y^{5}} \) simplifies to \( y^{2} \sqrt{y} \). The result is \( \frac{2 \sqrt{14} x^{2} \sqrt{x} y^{2} \sqrt{y}}{\sqrt{7} \sqrt{x} \sqrt{y}} \).

Step 3: Divide each pair of terms separately

Next, divide each pair of terms separately - one for the numbers and one for each variable. Dividing \( 2 \sqrt{14} \) by \( \sqrt{7} \) gives \( 2 \sqrt{2} \). Dividing \( x^{2} \sqrt{x} \) by \( \sqrt{x} \) gives \( x^{2} \), and dividing \( y^{2} \sqrt{y} \) by \( \sqrt{y} \) gives \( y^{2} \). Therefore, the simplified expression is \( 2 \sqrt{2} x^{2} y^{2} \).

Key Concepts

Understanding Square Roots

Square roots are a fundamental concept in mathematics and are pivotal for simplifying radical expressions. When we take the square root of a number, we're looking for a value that, when multiplied by itself, gives us the original number. For example, the square root of 25 is 5, because \(5 \times 5 = 25\).

Sometimes, you might encounter square roots of non-perfect squares, like \(\sqrt{56}\). For these cases, it's essential to break them down into products of two numbers where one is a perfect square, such as \(4 \times 14\). Hence, \(\sqrt{56} = \sqrt{4} \times \sqrt{14} = 2\sqrt{14}\).

When dealing with variables, the same principle applies. For instance, \(\sqrt{x^5}\) can be expressed as \(x^2\sqrt{x}\), since \(x^4\) is a perfect square \((x^2)^2\) and leaves \(\sqrt{x}\) as the remainder. Simplifying square roots is about recognizing these compositions to express them as simply as possible.

Division of Radicals Simplified

Division of radicals can seem tricky at first, but it's all about breaking down each component and simplifying them. When you have a division of square roots, such as \( \frac{\sqrt{56}}{\sqrt{7}} \), you can apply the property of radicals that allows you to combine them into one square root: \( \sqrt{\frac{56}{7}}\).

Continuing with this example, simplifying \( \frac{56}{7} \) becomes \(8\), resulting in \(\sqrt{8}\). Then, \(\sqrt{8}\) simplifies further to \(2\sqrt{2}\) because \(8 = 4 \times 2\) and \(\sqrt{4} = 2\).

With variables, the same principle helps. Divide the exponents by applying \( \frac{x^a}{x^b} = x^{a-b} \). In practice, for example, dividing \(x^2\sqrt{x}\) by \(\sqrt{x}\), results in just \(x^2\) because \((x^2 \times x^{1/2}) \div x^{1/2}\) ultimately resolves to \(x^{2}\). Remembering these basic properties will make dividing radicals straightforward and intuitive.

Simplification Processes Explained

Simplification processes, especially for expressions involving radicals, are systematic ways of reducing the complexity of the problem. The main goal of simplification is to present the expression in its most concise and understandable form. Here’s a simple breakdown of steps to follow:

• Factor numbers and variables under the square root to identify perfect squares.
• Rewrite the expression to isolate and simplify these perfect squares.
• Use fundamental algebraic principles, such as combining like terms and simplifying fractions.

For instance, in the expression \( \frac{2 \sqrt{14} x^{2} \sqrt{x} y^{2} \sqrt{y}}{\sqrt{7} \sqrt{x} \sqrt{y}} \), each part is simplified individually by canceling common terms. Dividing \(2\sqrt{14}\) by \(\sqrt{7}\) simplifies to \(2\sqrt{2}\). Likewise, dividing the variable parts results in \(x^2\) and \(y^2\), by eliminating the like \(\sqrt{x}\) and \(\sqrt{y}\) terms.

These steps, when mastered, allow you to quickly and efficiently break down complex radical expressions into their simplest forms.