Problem 343 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{\sqrt{26 y^{7}}}{\sqrt{2 y}} $$

Step-by-Step Solution

Answer

The simplified form is \( y^3 \sqrt{13} \).

Step 1: Combine the Radicals

Using the property of radicals that says \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \), combine the radicals into one. We get: \[ \frac{\sqrt{26 y^{7}}}{\sqrt{2 y}} = \sqrt{\frac{26 y^{7}}{2 y}} \]

Step 2: Simplify the Expression Inside the Radical

Divide the numerator by the denominator inside the radical: \[ \frac{26 y^{7}}{2 y} = \frac{26}{2} \cdot \frac{y^{7}}{y} = 13 y^{6} \] Thus, we get: \[ \sqrt{\frac{26 y^{7}}{2 y}} = \sqrt{13 y^{6}} \]

Step 3: Simplify the Radical Expression

Recognize that \(y^6\) is a perfect square, so we can take the square root of it: \[ \sqrt{13 y^{6}} = \sqrt{13} \cdot \sqrt{y^6} \] Combining these, we get: \[ \sqrt{13} \cdot y^{3} \] Thus, the simplified form is: \[ y^3 \sqrt{13} \]

Key Concepts

Combining Radicals

Combining radicals is a critical step in simplifying expressions that involve root functions. When you have a fraction with both the numerator and the denominator under a square root, you can combine them into a single radical. For example, in the exercise, \[ \frac{\sqrt{26 y^{7}}}{\sqrt{2 y}} \] becomes \[ \sqrt{\frac{26 y^{7}}{2 y}} \].
This transformation is possible due to the property of radicals:
\[ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \].
By combining the radicals, you simplify the expression under a single radical, making it easier to further simplify.

Properties of Radicals

Understanding the properties of radicals helps you simplify expressions effectively. Here are some key properties:

• \[ \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \]: This shows how you can break a radical into smaller factors.
• \[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \]: This is useful when you need to combine or separate radicals.
• \[ \sqrt{a^2} = |a| \]: This is fundamental for taking the square root of a squared term.

Using our example, when you combine the radicals, the next step involves simplifying the fraction inside the radical:
\[ \frac{26 y^{7}}{2 y} = \frac{26}{2} \cdot \frac{y^{7}}{y} = 13 y^{6} \].
These properties help you reduce the expression under the radical, leading to easier manipulation and simplification.

Simplifying Expressions

Simplifying expressions involves a few crucial steps to ensure the expression is in its simplest form. Begin by using properties of radicals to combine or break them apart. Next, handle any coefficients and variables within the radicals. In our example:

• First, we combined the radicals:
  \[ \frac{\sqrt{26 y^{7}}}{\sqrt{2 y}} = \sqrt{\frac{26 y^{7}}{2 y}} \].
• Then, we simplified the fraction inside the radical:
  \[ \frac{26}{2} \cdot \frac{y^{7}}{y} = 13 y^{6} \].
• Finally, recognizing that
  \[ y^6 \] is a perfect square, we can find its square root:
  \[ \sqrt{13 y^{6}} = \sqrt{13} \cdot \sqrt{y^6} = \sqrt{13} \cdot y^{3} \].
  Thus, \[ \sqrt{13} \cdot y^{3} \] is the simplified form.

Follow these steps for any radical expression, and you'll find simplifying becomes straightforward and manageable!