9514 1404 393
Answer:
2∛7
Step-by-step explanation:
Putting the expression in simplest form will factor cubes from under the radical.
![\displaystyle\sqrt[3]{56}=\sqrt[3]{2^3\cdot7}=\boxed{2\sqrt[3]{7}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csqrt%5B3%5D%7B56%7D%3D%5Csqrt%5B3%5D%7B2%5E3%5Ccdot7%7D%3D%5Cboxed%7B2%5Csqrt%5B3%5D%7B7%7D%7D)
Answer:
is proved for the sum of pth, qth and rth terms of an arithmetic progression are a, b,and c respectively.
Step-by-step explanation:
Given that the sum of pth, qth and rth terms of an arithmetic progression are a, b and c respectively.
First term of given arithmetic progression is A
and common difference is D
ie.,
and common difference=D
The nth term can be written as
![a_{n}=A+(n-1)D](https://tex.z-dn.net/?f=a_%7Bn%7D%3DA%2B%28n-1%29D)
pth term of given arithmetic progression is a
![a_{p}=A+(p-1)D=a](https://tex.z-dn.net/?f=a_%7Bp%7D%3DA%2B%28p-1%29D%3Da)
qth term of given arithmetic progression is b
and
rth term of given arithmetic progression is c
![a_{r}=A+(r-1)D=c](https://tex.z-dn.net/?f=a_%7Br%7D%3DA%2B%28r-1%29D%3Dc)
We have to prove that
![\frac{a}{p}\times (q-r)+\frac{b}{q}\times (r-p)+\frac{c}{r}\times (p-q)=0](https://tex.z-dn.net/?f=%5Cfrac%7Ba%7D%7Bp%7D%5Ctimes%20%28q-r%29%2B%5Cfrac%7Bb%7D%7Bq%7D%5Ctimes%20%28r-p%29%2B%5Cfrac%7Bc%7D%7Br%7D%5Ctimes%20%28p-q%29%3D0)
Now to prove LHS=RHS
Now take LHS
![\frac{a}{p}\times (q-r)+\frac{b}{q}\times (r-p)+\frac{c}{r}\times (p-q)](https://tex.z-dn.net/?f=%5Cfrac%7Ba%7D%7Bp%7D%5Ctimes%20%28q-r%29%2B%5Cfrac%7Bb%7D%7Bq%7D%5Ctimes%20%28r-p%29%2B%5Cfrac%7Bc%7D%7Br%7D%5Ctimes%20%28p-q%29)
![=\frac{A+(p-1)D}{p}\times (q-r)+\frac{A+(q-1)D}{q}\times (r-p)+\frac{A+(r-1)D}{r}\times (p-q)](https://tex.z-dn.net/?f=%3D%5Cfrac%7BA%2B%28p-1%29D%7D%7Bp%7D%5Ctimes%20%28q-r%29%2B%5Cfrac%7BA%2B%28q-1%29D%7D%7Bq%7D%5Ctimes%20%28r-p%29%2B%5Cfrac%7BA%2B%28r-1%29D%7D%7Br%7D%5Ctimes%20%28p-q%29)
![=\frac{A+pD-D}{p}\times (q-r)+\frac{A+qD-D}{q}\times (r-p)+\frac{A+rD-D}{r}\times (p-q)](https://tex.z-dn.net/?f=%3D%5Cfrac%7BA%2BpD-D%7D%7Bp%7D%5Ctimes%20%28q-r%29%2B%5Cfrac%7BA%2BqD-D%7D%7Bq%7D%5Ctimes%20%28r-p%29%2B%5Cfrac%7BA%2BrD-D%7D%7Br%7D%5Ctimes%20%28p-q%29)
![=\frac{Aq+pqD-Dq-Ar-prD+rD}{p}+\frac{Ar+rqD-Dr-Ap-pqD+pD}{q}+\frac{Ap+prD-Dp-Aq-qrD+qD}{r}](https://tex.z-dn.net/?f=%3D%5Cfrac%7BAq%2BpqD-Dq-Ar-prD%2BrD%7D%7Bp%7D%2B%5Cfrac%7BAr%2BrqD-Dr-Ap-pqD%2BpD%7D%7Bq%7D%2B%5Cfrac%7BAp%2BprD-Dp-Aq-qrD%2BqD%7D%7Br%7D)
![=\frac{[Aq+pqD-Dq-Ar-prD+rD]\times qr+[Ar+rqD-Dr-Ap-pqD+pD]\times pr+[Ap+prD-Dp-Aq-qrD+qD]\times pq}{pqr}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%5BAq%2BpqD-Dq-Ar-prD%2BrD%5D%5Ctimes%20qr%2B%5BAr%2BrqD-Dr-Ap-pqD%2BpD%5D%5Ctimes%20pr%2B%5BAp%2BprD-Dp-Aq-qrD%2BqD%5D%5Ctimes%20pq%7D%7Bpqr%7D)
![=\frac{Arq^{2}+pq^{2} rD-Dq^{2} r-Aqr^{2}-pqr^{2} D+qr^{2} D+Apr^{2}+pr^{2} qD-pDr^{2} -Ap^{2}r-p^{2} rqD+p^{2} rD+Ap^{2} q+p^{2} qrD-Dp^{2} q-Aq^{2} p-q^{2} prD+q^{2}pD}{pqr}](https://tex.z-dn.net/?f=%3D%5Cfrac%7BArq%5E%7B2%7D%2Bpq%5E%7B2%7D%20rD-Dq%5E%7B2%7D%20r-Aqr%5E%7B2%7D-pqr%5E%7B2%7D%20D%2Bqr%5E%7B2%7D%20D%2BApr%5E%7B2%7D%2Bpr%5E%7B2%7D%20qD-pDr%5E%7B2%7D%20-Ap%5E%7B2%7Dr-p%5E%7B2%7D%20rqD%2Bp%5E%7B2%7D%20rD%2BAp%5E%7B2%7D%20q%2Bp%5E%7B2%7D%20qrD-Dp%5E%7B2%7D%20q-Aq%5E%7B2%7D%20p-q%5E%7B2%7D%20prD%2Bq%5E%7B2%7DpD%7D%7Bpqr%7D)
![=\frac{Arq^{2}-Dq^{2}r-Aqr^{2}+qr^{2}D+Apr^{2}-pDr^{2}-Ap^{2}r+p^{2}rD+Ap^{2}q-Dp^{2}q-Aq^{2}p+q^{2}pD}{pqr}](https://tex.z-dn.net/?f=%3D%5Cfrac%7BArq%5E%7B2%7D-Dq%5E%7B2%7Dr-Aqr%5E%7B2%7D%2Bqr%5E%7B2%7DD%2BApr%5E%7B2%7D-pDr%5E%7B2%7D-Ap%5E%7B2%7Dr%2Bp%5E%7B2%7DrD%2BAp%5E%7B2%7Dq-Dp%5E%7B2%7Dq-Aq%5E%7B2%7Dp%2Bq%5E%7B2%7DpD%7D%7Bpqr%7D)
![=\frac{Arq^{2}-Dq^{2}r-Aqr^{2}+qr^{2}D+Apr^{2} -pDr^{2}-Ap^{2}r+p^{2}rD+Ap^{2}q-Dp^{2}q-Aq^{2}p+q^{2}pD}{pqr}](https://tex.z-dn.net/?f=%3D%5Cfrac%7BArq%5E%7B2%7D-Dq%5E%7B2%7Dr-Aqr%5E%7B2%7D%2Bqr%5E%7B2%7DD%2BApr%5E%7B2%7D%20-pDr%5E%7B2%7D-Ap%5E%7B2%7Dr%2Bp%5E%7B2%7DrD%2BAp%5E%7B2%7Dq-Dp%5E%7B2%7Dq-Aq%5E%7B2%7Dp%2Bq%5E%7B2%7DpD%7D%7Bpqr%7D)
![\neq 0](https://tex.z-dn.net/?f=%5Cneq%200)
ie., ![RHS\neq 0](https://tex.z-dn.net/?f=RHS%5Cneq%200)
Therefore
ie.,
Hence proved
To find y you need to insert x into the equation. The equation is y=2x+2, so insert the x on the table to find the y.
y=2(-4)+2=-6
y=2(-2)+2=-2
y=2(0)+2=2
y=2(1)+2=4
y=2(3)+2=8
These follow the equation's rule so it is correct.
Two plot the values of x and y you need to put it in (x,y) then plot it. Then you can check it by using the equation, y=mx+b, the b is the y intercept, so check to see if it starts there, and then check and see if it followings the slope
(-4, -6), (-2, -2), (0, 2), (1, 4), (3, 8)
Hope this helps, now you know the answer and how to do it. HAVE A BLESSED AND WONDERFUL DAY! As well as a great rest of Black History Month! :-)
- Cutiepatutie ☺❀❤
Pairs of numbers between 1 and 6 that add up to 7: 1+6, 2+5, 3+4, 4+3, 5+2, 6+1
Pairs of numbers between 1 and 6 that add up to 8: 2+6, 3+5, 4+4, 5+3, 6+2
Pairs of numbers between 1 and 6 that add up to 9: 3+6, 4+5, 5+4, 6+3
6 for 7, 5 for 8, 4 for 9. 15 in all