Problem 330 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\frac{\sqrt{10 y^{3}}}{\sqrt{5 y}}\) (b) \(\frac{\sqrt{108 n^{7}}}{\sqrt{243 n^{3}}}\)

Step-by-Step Solution

Answer

(a) \( \sqrt{2} y\), (b) \( \frac{2 n^{2}}{3}\)

Step 1: Simplify the Radicals

(a) Consider the expression \(\frac{\sqrt{10 y^{3}}}{\sqrt{5 y}}\). Apply the property of radicals \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), we get: \(\frac{\sqrt{10 y^{3}}}{\sqrt{5 y}} = \sqrt{\frac{10 y^{3}}{5 y}}\)

Step 2: Simplify the Fraction Inside the Radical

(a) Simplify \(\frac{10 y^{3}}{5 y}\) by dividing both the numerator and the denominator by 5: \(\frac{10 y^{3}}{5 y} = 2 y^{2}\). So, we have \(\sqrt{2 y^{2}}\).

Step 3: Simplify the Radical Expression

(a) Simplify \(\sqrt{2 y^{2}}\): we can separate it into \(\sqrt{2} \times \sqrt{y^{2}} \). Since \(\sqrt{y^{2}} = y\): the result is \(\sqrt{2} y\).

Step 4: Simplify the Radicals for Part (b)

(b) Consider the expression \(\frac{\sqrt{108 n^{7}}}{\sqrt{243 n^{3}}}\). Like in part (a), we can apply property: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), \(\frac{\sqrt{108 n^{7}}}{\sqrt{243 n^{3}}} = \sqrt{\frac{108 n^{7}}{243 n^{3}}}\).

Step 5: Simplify the Fraction Inside the Radical for Part (b)

(b) Simplify \(\frac{108 n^{7}}{243 n^{3}}\). First, simplify \(\frac{108}{243}\): by dividing both numerator and denominator by 27, we get \(\frac{108}{243} = \frac{4}{9}\). Then, simplify \(\frac{n^{7}}{n^{3}} = n^{4}\). So we get \(\sqrt{\frac{4 n^{4}}{9}}\).

Step 6: Simplify the Radical Expression for Part (b)

(b) Separate \(\sqrt{\frac{4 n^{4}}{9}} = \frac{\sqrt{4 n^{4}}}{\sqrt{9}}\). Since \(^4\) is a perfect square, we get \(^2\) similarly for 4 and 9, we get 2 and 3 respectively. Combine to obtain \(\frac{2 n^{2}}{3}\).

Key Concepts

radical expressions

A radical expression involves roots such as square roots, cube roots, etc. The most common root is the square root, denoted by the radical symbol \( \sqrt{} \). For example, \( \sqrt{a} \) represents the square root of \ 'a' \.
To simplify radical expressions, follow these steps:

• Combine like terms if possible.
• Use the property of radicals: \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) to break down complex expressions.
• Factor the number inside the radical to its prime factors.
• Simplify the radicals wherever possible.

For example, consider simplifying \( \sqrt{2y^2} \). We separate it as \( \sqrt{2} \cdot \sqrt{y^2} \). Since \( \sqrt{y^2} = y \), the expression simplifies to \( \sqrt{2} y \). By understanding these basic principles, you can handle more complex radical expressions more easily.

simplifying fractions

Simplifying fractions is a vital skill in algebra. It involves reducing the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).
Let's break down the steps:

• Find the GCD of the numerator and the denominator.
• Divide both parts of the fraction by the GCD.

For instance, in the expression \( \frac{108}{243} \), we find the GCD, which is 27. Dividing both by 27, we get \( \frac{4}{9} \).
Another useful property in dealing with fractions and radicals is: \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\.
This property makes it easier to work with complex radial fractions, by turning the division into a single radical.

algebraic properties

Algebraic properties like the associative, commutative, and distributive properties are fundamental for manipulating mathematical expressions. Here are some key points:

• Associative Property: \( (a + b) + c = a + (b + c) \) — This property states that the grouping of numbers doesn’t affect their sum.
• Commutative Property: \( a + b = b + a \) — Order of numbers doesn’t affect their sum.
• Distributive Property: \( a(b + c) = ab + ac \) — This property allows you to distribute a multiplication over an addition.

These properties also extend to radicals. For example: \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \) follows both the commutative and associative properties.
Why is this useful? When simplifying radical expressions like \( \sqrt{10y^3} / \sqrt{5y} \), you apply both distributive and commutative properties to rephrase and further simplify the expression. Thus, these properties help streamline solving algebraic problems.