Please simplify the following expressions and show steps:   1.        `((4t)^-2 t^0)/(t^6 v^-8)`              2.        ` (p^-3)/(p^-2r^3)` 3.        ...

Please simplify the following expressions and show steps: 

 1.        `((4t)^-2 t^0)/(t^6 v^-8)`            

 2.        ` (p^-3)/(p^-2r^3)`

3.         `(5x^-7)^2`

Write a rule for the following functions:

x      -1, 0, 1, 2

y       1, 3, 9, 27

x    -1, 0, 1,     2

y     3, 1, 1/3, 1/9

Expert Answers
embizze eNotes educator| Certified Educator

(1) Simplify `(4t^(-2)t^0)/(t^6v^(-8)) ` :

We use the following properties of exponents: `a^(-m)=1/(a^m) ` ,`a^0=1 ` if `a !=0 ` , `a^m*a^n=a^(m+n) `

`((4t)^(-2)t^0)/(t^6v^(-8))=(4^(-2)t^(-2)t^0)/(t^6v^(-8)) `

`=1/4^2*(t^(-2)t^0)/t^6*1/v^(-8) `

`((4t)^(-2)t^0)/(t^6v^(-8))=1/16v^8/t^8 `

(2) Simplify `p^(-3)/(p^(-2)*r^3) `

`p^(-3)/(p^(-2)r^3)=(p^(-3)*p^2)/r^3=p^(-1)/r^3=1/(p*r^3) `

Alternatively you can use `a^m/a^n=a^(m-n) ` ; `p^(-3)/p^(-2)=p^(-3-(-2))=p^(-1)=1/p `

(3) Simplify `(5x^(-7))^2 ` : Use the property `(ab)^m=a^mb^m `

`(5x^(-7))^2=5^2(x^(-7))^2 `   Use the property `(a^m)^n=a^(m*n) `

`=25x^(-7*2)=25x^(-14)=25/x^14 `

For the last two problems, note that there are an infinite number of functions that will contain the four given points. Since the previous questions dealt with exponents, we will give an answer involving powers.

(a) (-1,1),(0,3),(1,9),(2,27) Note that the y-values are powers of 3 `(3^0=1,3^1=3,3^2=9,3^3=27) `

One rule/function is `y=3^(x-1) ` (Note that `y=4/3x^3+2x^2+8/3x+3 ` will also give the same four points)

(b) `(-1,3),(0,1),(1,1/3),(2,1/9) ` Again note that the y-values are powers of 3: `(3^1=3,3^0=1,3^(-1)=1/3,3^(-2)=1/9) `

One rule/function is `y=3^(-x) ` (Note that `y=-4/27x^3+2/3x^2-32/27x+1 ` will also yield the same 4 points.)

oldnick | Student

1)  

`((4t)^(-2)t^0)/(t^6v^(-8))=(1/16 t^(-2))/(t^6v^(-8))=` `1/16 t^(-2) t^(-6) v^(-(-8))=` `1/16t^(-8)v^8=v^8/(16t^8)=1/16 (v/t)^8`

2)

`p^(-3)/p^(-2) r^3=p^(-3)p^(-(-2))r^(-3)=p^(-3) p^2 r^(-3)=` `p^(-3+2)r^(-3)=`

`=p^(-1)r^(-3)=1/(pr^3)`