Combine and simplify the following radical expression:
$(3 \sqrt{y})^2$
Answer (2)
The expression ( 3 y ) 2 simplifies to 9 y by applying the power of a product rule and simplifying both the constant and the radical. The steps involve squaring the number and the square root separately and then combining the results. Therefore, the final answer is 9 y .
;
Apply the power of a product rule: $(3
√y)^2 = 3^2 * (
√y)^2$.
Simplify 3 2 to 9 .
Simplify $(
√y)^2 t o y$.
Combine the results to get the final simplified expression: 9 y .
Explanation
Understanding the Problem We are given the expression $(3
√y)^2$ to simplify. Our goal is to combine and simplify this expression using the properties of exponents and radicals.
Applying the Power of a Product Rule Let's break down the expression step by step. First, we apply the power of a product rule, which states that ( ab ) n = a n b n . In our case, this means: ( 3√ y ) 2 = 3 2 ( √ y ) 2
Simplifying the Constant Term Now, let's simplify each part separately. We know that 3 2 = 3 ∗ 3 = 9 . So, we have: 3 2 = 9
Simplifying the Radical Term Next, we simplify $(
√y)^2 . T h es q u a reroo t o f an u mb er , w h e n s q u a re d , s im pl y g i v es u s t h eor i g ina l n u mb er ( a ss u min g y i s n o n − n e g a t i v e ) . T h ere f ore : ( √ y ) 2 = y $
Combining the Results Now, we combine the simplified parts: 9 ∗ y = 9 y
Final Answer So, the simplified expression is 9 y .
Examples
Imagine you are calculating the area of a square where the side length is $3
√y . T h e a re a o f a s q u a re i s t h es i d e l e n g t h s q u a re d , w hi c h w o u l d b e (3
√y)^2$. Simplifying this expression to 9 y gives you a direct formula for the area in terms of y . For example, if y = 4 , the area would be 9 ∗ 4 = 36 square units. This type of simplification is useful in various geometric and algebraic problems.
