Problem 111 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{y^{4}}{y^{8}}} $$

Step-by-Step Solution

Answer

\frac{1}{y^2}

Step 1: Write the expression under a single square root

Rewrite the expression \(\frac{y^4}{y^8}\) under a single square root: \(\frac{y^4}{y^8}\)

Step 2: Simplify the fraction inside the square root

Subtract the exponent of the denominator from the exponent of the numerator: \(\frac{y^4}{y^8} = y^{4-8} = y^{-4}\)

Step 3: Apply the square root to the simplified expression

Take the square root of \(y^{-4}\): \(\root{y^{-4}} = y^{-2}\)

Step 4: Simplify the final expression

Using the property of negative exponents, rewrite \((y^{-2})\) as \(\frac{1}{y^2}\). So, \(\root{\frac{y^4}{y^8}} = \frac{1}{y^2}\)

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. Here, we have a square root, specifically \(\root{...}\) notation. Sqrt (square root) means finding a number that, when multiplied by itself, gives the original number.
In the given exercise, we start with the expression \(\root{\frac{y^{4}}{y^{8}}}\). To simplify it, we need first to understand the concept of radicals and how to work under the root symbol. This often involves simplifying what's inside the root first.

Exponent Rules

Exponents indicate how many times a number (the base) is multiplied by itself. Important rules include:

• \textbf{Multiplication Rule:} \(a^m \times a^n = a^{m+n}\)
• \textbf{Division Rule:} \(a^m / a^n = a^{m-n}\)
• \textbf{Power Rule:} \( (a^m)^n = a^{mn} \)

In the exercise, we have \( \frac{y^4}{y^8} \). Using the division rule, we get \( y^{4-8} = y^{-4} \). Exponent rules help us move the problem forward toward simplification.

Simplifying Fractions

Simplifying fractions sometimes means reducing to the lowest terms or, in the case of exponents, manipulating them based on the rules. For exponents:
\( \frac{y^4}{y^8} \) translates to \( y^{4-8} \)
. This gives \( y^{-4} \). You simplify by cancelling common factors or applying exponent rules.
In our problem, simplifying inside the root brings us to a simpler expression before we take the square root.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive power. For example, \( a^{-n} = \frac{1}{a^n} \).
In the exercise, we get \( y^{-4} \). To simplify this inside the square root:
First, the square root of \( y^{-4} = y^{-2} \). Finally, converting the negative exponent: \( y^{-2} = \frac{1}{y^2} \). Thus, \( \root{\frac{y^4}{y^8}} = \frac{1}{y^2} \). Negative exponents are crucial for transforming and simplifying radical expressions.