Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ 3 \sqrt{20 y^{5}} $$

Step-by-Step Solution

Answer

The simplest radical form is \(6y^2 \sqrt{5y}\).

Step 1: Factor the Radicand

First, we need to factor the expression inside the radical, which is \(20y^5\). The number \(20\) can be factored into prime numbers as \(4 \times 5\), and \(4\) can be further factored into \(2^2\). So, \(20 = 2^2 \times 5\). Therefore, \(20y^5\) becomes \((2^2 \times 5) y^5\).

Step 2: Simplify the Square Root

The expression inside the radical is now \(\sqrt{2^2 \times 5 \times y^5}\). The square root of \(2^2\) is \(2\) (since \(\sqrt{2^2} = 2\)). Also, for \(y^5\), we can extract \(y^2\) (as \(y^4\)) since it's a perfect square, leaving us with \(y\) inside. So, \(y^5 = y^4 \times y = (y^2)^2 \times y\). Thus, \(\sqrt{y^5} = y^2 \sqrt{y}\). Therefore, \(\sqrt{20y^5} = 2 y^2 \sqrt{5y}\).

Step 3: Write in Simplest Radical Form

Finally, multiply by the coefficient outside the square root. The coefficient is \(3\), and the square root expression we simplified is \(2y^2\sqrt{5y}\). So, multiplying them together gives us: \(3 \times 2y^2 \times \sqrt{5y} = 6y^2 \sqrt{5y}\).

Key Concepts

Square Roots

Square roots are a fundamental part of mathematics, frequently encountered in problems involving radicals and algebraic expressions. When you take the square root of a number or an algebraic expression, you are essentially looking for a value that, when multiplied by itself, yields the original number or expression. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. It's denoted by the symbol \( \sqrt{} \).

Understanding square roots doesn't just apply to numbers, but also to algebraic expressions. You can split a square root across a product, such as \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This property is particularly useful in simplifying expressions like \( \sqrt{20y^5} \), where it helps to break it down into more manageable parts.

Another key point to remember is that square roots are only real numbers when dealing with perfect squares or positive numbers. This is because a square root of a negative number involves imaginary numbers, which are denoted differently in more advanced math.

Radicals

In algebra, radicals are expressions that include a root symbol, such as a square root or cube root. They can often seem complex, but understanding the basics of how to handle them can simplify your work with these expressions.

The main task when working with radicals like \( \sqrt{20y^5} \) is to express them in their simplest or simplest radical form. This involves breaking down the expression under the radical into its prime factors, making it easier to identify and pull out perfect squares or other perfect powers.

• To simplify a radical, factor the number or expression under the root to its simplest components.
• Identify perfect squares or cubes that can be taken out of the square root or cube root, respectively.
• When simplified, the radical may still contain some components that cannot be simplified into whole numbers, which remain under the radical sign.

So, for \( \sqrt{20y^5} \), recognizing the parts \( 2^2 \) and \( y^4 \) as perfect squares allows us to extract them, thus simplifying the whole radical expression as much as possible.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like +, –, ×, ÷). They represent a well-defined mathematical phrase which can be simplified or manipulated using various algebraic rules.

In the context of radicals, algebraic expressions often involve simplifying terms inside the radical, like the expression \( 20y^5 \) we started with. Simplifying algebraic expressions involving radicals means using properties of exponents along with factoring techniques to identify simplifiable parts.

When dealing with expressions like \( \sqrt{20y^5} \), keep the following tips in mind:

• Factor numerical coefficients and powers of variables separately, such as breaking 20 into \( 2^2 \times 5 \) and \( y^5 \) into \( y^4 \times y \).
• Apply the same rules you learned with numbers to variables — treat \( y^4 \) as \( (y^2)^2 \), making it easier to take square roots.
• Be wary of maintaining the conditions specified in the problem, such as non-negative conditions for even index radicands and non-zero conditions for denominators.

Understanding these properties helps you manage algebraic expressions confidently, ensuring that you can simplify complex problems into manageable, simplified radical forms.