Simplify the following: $\left(4 x^6 y^5\right)^2$ = $\square$ $\sqrt{x}$
Use the $\square$ button to enter your answer. Input your answer WITHOUT spaces and the variables in ALP
Answer (1)
Expand the expression: ( 4 x 6 y 5 ) 2 = 16 x 12 y 10 .
Rewrite x 12 as x 2 23 x .
Express the simplified form as 16 x 2 23 y 10 x .
The final answer is 16 x 2 23 y 10 .
Explanation
Understanding the Problem We are given the expression ( 4 x 6 y 5 ) 2 and we want to simplify it in the form □ x . The simplified expression should be in terms of x and y without spaces, and the variables should be in alphabetical order.
Expanding the Expression First, we need to expand the given expression. When we raise a product to a power, we raise each factor to that power. So, we have: ( 4 x 6 y 5 ) 2 = 4 2 ∗ ( x 6 ) 2 ∗ ( y 5 ) 2
Simplifying the Terms Now, we simplify each term: 4 2 = 16 ( x 6 ) 2 = x 6 ∗ 2 = x 12 ( y 5 ) 2 = y 5 ∗ 2 = y 10 So, the expression becomes: 16 x 12 y 10
Factoring out the Square Root We want to express this in the form □ x . This means we need to factor out x from x 12 . Recall that x = x 2 1 = x 0.5 .
We can rewrite x 12 as x 11.5 ∗ x 0.5 = x 2 23 ∗ x .
So, we have: 16 x 12 y 10 = 16 ∗ x 2 23 ∗ x ∗ y 10 = 16 x 2 23 y 10 x
Final Answer Thus, the expression in the required form is 16 x 2 23 y 10 x . Therefore, the missing part is 16 x 2 23 y 10 .
Examples
This type of simplification is useful in physics when dealing with equations involving areas or volumes that scale with powers of variables. For instance, if you have a square area that depends on x 6 and y 5 , squaring the area would result in an expression similar to the one we simplified. Understanding how to manipulate these expressions helps in calculating how changes in x and y affect the overall area.
