What is the range of possible sizes for side x?

Step-by-step explanation:

You use the triangle inequality theorem for this.  That theorem states that if the sides are lengths a, b, and c, then

a + b > c,

b + c > a, and

a + c > b.

Let a = 5.5, b = 3.2, and c = x.

5.5 + 3.2 > x, so 8.7 > x (aka x < 8.7)

3.2 + x > 5.5, so x > 2.3 and

5.5 + x > 3.2, so x > -2.3

Since measurements of length and distance will never be negative, our range of values for x is

2.3 < x < 8.7

If x were to be shorter than 2.3, the triangle wouldn't "close" up; likewise if one side were to be longer than 8.7...another side wouldn't "close".

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation: