Simplify $\sqrt{48 n^5}$.
The simplified expression is $\square$.
Answer (2)
Factor the constant: 48 = 2 4 × 3 .
Rewrite the variable term: n 5 = n 4 × n .
Separate and simplify perfect squares: 2 4 × n 4 × 3 × n = 2 2 n 2 3 n .
The simplified expression is 4 n 2 3 n .
Explanation
Understanding the Problem We are asked to simplify the expression 48 n 5 . This involves finding perfect square factors within the square root and simplifying them.
Factoring the Constant First, we can factor 48 into its prime factors. We have 48 = 16 × 3 = 2 4 × 3 . So, we can rewrite the expression as 2 4 × 3 × n 5 .
Factoring the Variable Term Next, we can rewrite n 5 as n 4 × n . This allows us to identify perfect square factors involving n . Now our expression is 2 4 × 3 × n 4 × n .
Separating Perfect Squares Now, we separate the perfect squares: 2 4 × n 4 × 3 × n = 2 4 × n 4 × 3 n .
Simplifying Simplifying the perfect squares, we have 2 2 × n 2 × 3 n = 4 n 2 3 n .
Final Answer Therefore, the simplified expression is 4 n 2 3 n .
Examples
Square root simplification is used in various fields, such as physics and engineering, to simplify calculations involving distances, areas, and volumes. For example, when calculating the length of the diagonal of a square with side length s , we use the Pythagorean theorem to find that the diagonal is s 2 + s 2 = 2 s 2 = s 2 . Simplifying square roots allows for easier manipulation and understanding of these quantities.
The simplified expression for 48 n 5 is 4 n 2 3 n . This result is obtained by factoring 48 and n 5 , separating the perfect squares, and simplifying. Following these steps helps clarify the simplification process for square roots with polynomial terms.
;
