Select the correct answer.
Which expression is equivalent to the given expression?
[tex]$8 \sqrt{6}$[/tex]
A. [tex]$\sqrt{48}$[/tex]
B. [tex]$\sqrt{96}$[/tex]
C. [tex]$\sqrt{576}$[/tex]
D. [tex]$\sqrt{384}$[/tex]
Answer (1)
Rewrite 8 6 as 8 2 ⋅ 6 .
Calculate 8 2 = 64 .
Multiply 64 ⋅ 6 = 384 .
Therefore, 8 6 = 384 , and the correct answer is D: 384 .
Explanation
Understanding the problem We are given the expression 8 6 and asked to find an equivalent expression among the choices A. 48 , B. 96 , C. 576 , D. 384
Solution Strategy The key idea is to rewrite 8 6 in the form x for some x, and then compare x with 48, 96, 576, and 384.
Rewriting the expression To rewrite 8 6 in the form x , we need to move the 8 inside the square root. Remember that a b = a 2 b . So, we have
8 6 = 8 2 ⋅ 6
Calculating the value Now, we calculate 8 2 ⋅ 6 . We know that 8 2 = 64 , so we have
8 2 ⋅ 6 = 64 ⋅ 6
Finding the equivalent expression Next, we multiply 64 by 6:
64 ⋅ 6 = 384
Therefore, 8 6 = 384 .
Selecting the correct answer Comparing 384 with the given choices, we see that option D is the correct answer.
Examples
Understanding how to simplify and manipulate radical expressions like 8 6 is useful in various fields, such as engineering and physics, where calculations involving lengths, areas, and volumes often require working with square roots. For example, when calculating the length of the diagonal of a rectangle or the distance between two points in a coordinate system, you might encounter expressions involving square roots that need simplification. Knowing how to rewrite these expressions can make calculations easier and more accurate. Imagine you are building a rectangular garden and need to calculate the length of a diagonal fence; this skill becomes essential for precise measurements and cost estimation.
