Problem 67 - Algebra 2 and Trigonometry

Question

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \left(25 x^{2} y\right)^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

\(5x\sqrt{y}\)

Step 1: Recognize the power of 1/2

The expression \((25x^{2}y)^{\frac{1}{2}}\) is a square root because raising something to the power of \(\frac{1}{2}\) is equivalent to taking the square root.

Step 2: Apply the radical expression to each term separately

To express the entire expression under one radical, rewrite \((25x^{2}y)^{\frac{1}{2}}\) as \(\sqrt{25x^{2}y}\). This allows each term under the square root to be simplified separately.

Step 3: Simplify each term under the square root

Calculate the square root of each component: \(\sqrt{25} = 5\), \(\sqrt{x^{2}} = x\), and \(y\) remains under the square root since \(y\) is not a perfect square. So, we have \(5x\sqrt{y}\).

Step 4: Combine the simplified terms into the final expression

Organize the simplified expression: the coefficient and fully simplified variables (in this instance, \(5x\)) multiplied by the remaining radical term \(\sqrt{y}\). This results in the simplest form being \(5x\sqrt{y}\).

Key Concepts

Square Roots

A square root is the opposite of squaring a number. It involves finding which number multiplied by itself will give you the original number. For instance, the square root of 25 is 5 because 5 times 5 equals 25.

In algebra, when we deal with variables, the concept remains the same. If you have a term like \(x^2\), its square root would simply be \(x\), assuming \(x\) is a positive number. This concept is particularly useful when working with expressions in different forms, such as radical expressions.

Square roots become even more important in dealing with fractional exponents. For example, an exponent of \(\frac{1}{2}\) indicates a square root; applying it to an expression like \((25x^2y)^\frac{1}{2}\) transforms it into a radical expression: \(\sqrt{25x^2y}\).

Simplifying Expressions

Simplifying expressions is all about making them easier to work with. The goal is to make an expression as straightforward as possible by combining like terms and simplifying radicals when applicable.

Let's take the expression \(\sqrt{25x^2y}\). To simplify this, you should consider each component separately. Here’s how you do it:

• Start with numbers: Simplify \(\sqrt{25}\) to get 5.
• Look at variables with even exponents: \(\sqrt{x^2}\) simplifies to \(x\).
• For variables like \(y\) that don’t simplify perfectly, leave them under the radical: \(\sqrt{y}\).

After simplifying each part, combine them back together to get \(5x\sqrt{y}\).

The process of simplifying ensures that the expression is in its simplest and most usable form, which is important for further calculations or solving equations.

Exponents

Exponents are a shorthand way of showing repeated multiplication. For example, \(x^2\) means \(x\) is multiplied by itself once. When dealing with exponents, you'll often find yourself working with laws that help simplify expressions.

One such exponent rule is the fractional exponent, which is another way of expressing roots. For instance, \((25x^2y)^\frac{1}{2}\) uses an exponent \(\frac{1}{2}\), indicating that the expression is equivalent to taking the square root of \(25x^2y\).

Understanding how to manipulate and simplify expressions involving exponents is key in algebra. It allows for easier manipulation of complex expressions and turning them into more manageable forms. Be sure to apply the rules of exponents carefully so you achieve the correct simplified form, making calculations smoother and more efficient.