Problem 245 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\quad(\sqrt{15 y})\left(\sqrt{5 y^{3}}\right)\) (b) \(\left(\sqrt{2 n^{2}}\right)\left(\sqrt{18 n^{3}}\right)\)

Step-by-Step Solution

Answer

For (a): \ \( 5 \sqrt{3} y^2 \). For (b): \ \( 6n^{5/2} \).

Step 1: Simplify the Square Roots for Part (a)

Simplify each square root individually: \ \( \sqrt{15y} \) and \( \sqrt{5y^3} \).

Step 2: Multiply the Expressions under the Square Roots - Part (a)

Combine the expressions under a single square root: \ \( \sqrt{15y} \cdot \sqrt{5y^3} = \sqrt{(15y)(5y^3)} \). This simplifies to \ \( \sqrt{75y^4} \).

Step 3: Simplify the Result - Part (a)

Simplify \ \( \sqrt{75y^4} \): \ \( \sqrt{75} \cdot \sqrt{y^4} = \sqrt{25 \cdot 3} \cdot y^2 = 5 \sqrt{3} y^2 \). This is the final simplified form.

Step 4: Simplify the Square Roots for Part (b)

Simplify each square root individually: \ \( \sqrt{2n^2} \) and \( \sqrt{18n^3} \).

Step 5: Multiply the Expressions under the Square Roots - Part (b)

Combine the expressions under a single square root: \ \( \sqrt{2n^2} \cdot \sqrt{18n^3} = \sqrt{(2n^2)(18n^3)} \). This simplifies to \ \( \sqrt{36n^5} \).

Step 6: Simplify the Result - Part (b)

Simplify \ \( \sqrt{36n^5} \): \ \( \sqrt{36} \cdot \sqrt{n^5} = 6 \cdot n^{2.5} = 6n^{5/2} \). This is the final simplified form.

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
It's used to express equations and relationships. In the context of simplifying square roots and expressions, algebra helps us understand and apply rules.
Key principles include:
- Variables: Symbols like x, y, and n that stand in for values.
- Constants: Specific numbers, like 2 or 5.
- Operations: Addition, subtraction, multiplication, and division.
When working with square roots, it's important to know how to manipulate and simplify these symbols and numbers accurately.

Multiplying Square Roots

Multiplying square roots involves a few straightforward steps.
Let's break it down using our exercise: \( (\sqrt{15 y})(\sqrt{5 y^{3}}) \).
1. **Simplify the individual square roots**: Simplify each part separately. For example, for \(\sqrt{15y} \) and \( \sqrt{5y^3} \). This doesn't change their values but may involve breaking them into simpler parts.
2. **Combine under one square root**: Multiply the expressions under one larger square root. This becomes \( \sqrt{(15y)(5y^3)} = \sqrt{75y^4} \).
3. **Simplify the result**: Break down the expression into simpler square roots. For example, \( \sqrt{75y^4} = \sqrt{75} \cdot \sqrt{y^4} = 5 \sqrt{3} y^2 \). Practicing these steps will make the process easier and more intuitive.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form.
This makes them easier to work with and understand. When simplifying square root expressions:
1. **Factorize and Extract Square Roots**: For instance, \( \sqrt{75} = \sqrt{25 \cdot 3} = 5 \sqrt{3} \) because 25 is a perfect square.
2. **Combine Like Terms**: Make sure to simplify all possible parts. For \( \sqrt{y^4} = y^2 \) because \( y^4 \) is a perfect square.
3. **Use Rational Exponents**: For division or complex forms, converting to rational exponents helps. For example, \( \sqrt{n^5} = n^{5/2} \). This approach is extremely helpful in understanding higher-level algebra problems and making complex calculations manageable.