Questions on Algebra: Exponents and operations on exponents answered by real tutors!

A retired woman has $80,000 to invest but needs to make $3,000 a year from the interest to meet certain living expenses.
One bond investment pays 15% annual interest. The rest of it she wants to put in a CD that pays 7%. 

If we let x be the amount the woman invests in the 15% bond, how much in dollars will she be able to invest in the CD? 
*************************************
She can NEVER invest any part of $80,000 in both instruments, simultaneously, to acquire $3,000 annually!

To secure annual interest income of $3,000, she'd just have to invest  in the
smaller-interest investment: the CD, which "pays" 7%. No FURTHER investment would be required. As a
result, $37,142.86 would remain uninvested. 

If investing in the bond instead, only $20,000 () would be needed to secure $3,000, annually, thereby
leaving $60,000 uninvested.

Bottom LINE: At these interest rates, she wouldn't need to invest the entire $80,000 in both the CD and the
bond. Either would suffice, with funds left over.

Where does the parentheses go in 10 + 5 times 4 to the power of 2 divided by 2 to the power of 3= 20
*******************
10 + 5 times 4 to the power of 2 divided by 2 to the power of 3 = 20

 =  =  = 

(4/3)to the 3rd power divided by 2-2/3

tried several different ways, I know the answer is 16/9 but don't know how to get it.

I tried.

64/27 x -4/3 then I flipped to -3/4

came up with 61/23
==================
You're trying to do too much. It's much easier doing it this way!
 =  =  =  =  

The 21th term is  C(44,20)*x^20 = .


The 22th term is  C(44,21)*x^21 = .


They are equal

     = .


It implies

     = x,

or

    x =  = .



ANSWER.  x = .

If you will rise your number 7 in consecutive degrees, you will find very soon, that the last 
digit is repeating.


This is not accidental - it is natural.


Indeed, every time you rise to the next degree, the last digit is a result of multiplication 
of the previous last digit by 7.

But the number of possible last digits is limited: it is only 10, because there are 10 digits,
from 0 to 9, in all.

Therefore inevitably, some last digit will repeat one of preceding last digits.

As soon as it will happen for the first time, the digits will be repeating cyclically.


Let's consider how it works in our case. I will not write whole many-digit numbers,
since there is no need in it. I only will write and track the last digit.


    degree k          1    2    3    4    5    6    7    8    9

    the last digit    7    9    3    1    7    9    3    1    7
    of  


You see that first time repeating happens at k = 5,  and after k=4 the cycle (7,9,3,1)  is repeating again and again.


So, for  ,  there are many repeating cycles of the length 4 - they are not interesting for us.


The number 155 gives the remainder  3  when divided by 4:  155 = 4*38 + 3.


It tells us that the sough last digit in the number    is the third term in the basic  cycle of four digits  {7, 9, 3, 1}.


From it, we conclude that the last digit of the number    is 3, since '3' is the third term of the cycle {7, 9, 3, 1}.