2)
a)
![\bf a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^{ n}} \qquad \qquad \sqrt[{ m}]{a^{ n}}\implies a^{\frac{{ n}}{{ m}}}\\\\ -------------------------------\\\\ (4x^5\cdot x^{\frac{1}{3}})+(2x^4\cdot x^{\frac{1}{3}})-(7x^3\cdot x^{\frac{1}{3}})+(3x^2\cdot x^{\frac{1}{3}})\\\\+(9x^1\cdot x^{\frac{1}{3}})-(1\cdot x^{\frac{1}{3}}) \\\\\\ 4x^{5+\frac{1}{3}}+2x^{4+\frac{1}{3}}-7x^{3+\frac{1}{3}}+9x^{1+\frac{1}{3}}-x^{\frac{1}{3}}](https://tex.z-dn.net/?f=%5Cbf%20a%5E%7B%5Cfrac%7B%7B%20n%7D%7D%7B%7B%20m%7D%7D%7D%20%5Cimplies%20%20%5Csqrt%5B%7B%20m%7D%5D%7Ba%5E%7B%20n%7D%7D%20%5Cqquad%20%5Cqquad%0A%5Csqrt%5B%7B%20m%7D%5D%7Ba%5E%7B%20n%7D%7D%5Cimplies%20a%5E%7B%5Cfrac%7B%7B%20n%7D%7D%7B%7B%20m%7D%7D%7D%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C%0A%284x%5E5%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%29%2B%282x%5E4%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%29-%287x%5E3%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%29%2B%283x%5E2%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%29%5C%5C%5C%5C%2B%289x%5E1%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%29-%281%5Ccdot%20x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%29%0A%5C%5C%5C%5C%5C%5C%0A4x%5E%7B5%2B%5Cfrac%7B1%7D%7B3%7D%7D%2B2x%5E%7B4%2B%5Cfrac%7B1%7D%7B3%7D%7D-7x%5E%7B3%2B%5Cfrac%7B1%7D%7B3%7D%7D%2B9x%5E%7B1%2B%5Cfrac%7B1%7D%7B3%7D%7D-x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D)
![\bf 4x^{\frac{16}{3}}+2x^{\frac{13}{3}}-7x^{\frac{10}{3}}+9x^{\frac{4}{3}}-x^{\frac{1}{3}} \\\\\\ 4\sqrt[3]{x^{16}}+2\sqrt[3]{x^{13}}-7\sqrt[3]{x^{10}}+9\sqrt[3]{x^4}-\sqrt[3]{x}](https://tex.z-dn.net/?f=%5Cbf%204x%5E%7B%5Cfrac%7B16%7D%7B3%7D%7D%2B2x%5E%7B%5Cfrac%7B13%7D%7B3%7D%7D-7x%5E%7B%5Cfrac%7B10%7D%7B3%7D%7D%2B9x%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D-x%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%0A%5C%5C%5C%5C%5C%5C%0A4%5Csqrt%5B3%5D%7Bx%5E%7B16%7D%7D%2B2%5Csqrt%5B3%5D%7Bx%5E%7B13%7D%7D-7%5Csqrt%5B3%5D%7Bx%5E%7B10%7D%7D%2B9%5Csqrt%5B3%5D%7Bx%5E4%7D-%5Csqrt%5B3%5D%7Bx%7D)
b)
![\bf 4x^{5-\frac{1}{3}}+2x^{4-\frac{1}{3}}-7x^{3-\frac{1}{3}}+9x^{1-\frac{1}{3}}-x^{-\frac{1}{3}} \\\\\\ 4x^{\frac{14}{3}}+2x^{\frac{11}{3}}-7x^{\frac{8}{3}}+9x^{\frac{2}{3}}-x^{-\frac{1}{3}} \\\\\\ 4\sqrt[3]{x^{14}}+2\sqrt[3]{x^{11}}-7\sqrt[3]{x^{8}}+9\sqrt[3]{x^{2}}-\frac{1}{\sqrt[3]{x}}](https://tex.z-dn.net/?f=%5Cbf%204x%5E%7B5-%5Cfrac%7B1%7D%7B3%7D%7D%2B2x%5E%7B4-%5Cfrac%7B1%7D%7B3%7D%7D-7x%5E%7B3-%5Cfrac%7B1%7D%7B3%7D%7D%2B9x%5E%7B1-%5Cfrac%7B1%7D%7B3%7D%7D-x%5E%7B-%5Cfrac%7B1%7D%7B3%7D%7D%0A%5C%5C%5C%5C%5C%5C%0A4x%5E%7B%5Cfrac%7B14%7D%7B3%7D%7D%2B2x%5E%7B%5Cfrac%7B11%7D%7B3%7D%7D-7x%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%2B9x%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D-x%5E%7B-%5Cfrac%7B1%7D%7B3%7D%7D%0A%5C%5C%5C%5C%5C%5C%0A4%5Csqrt%5B3%5D%7Bx%5E%7B14%7D%7D%2B2%5Csqrt%5B3%5D%7Bx%5E%7B11%7D%7D-7%5Csqrt%5B3%5D%7Bx%5E%7B8%7D%7D%2B9%5Csqrt%5B3%5D%7Bx%5E%7B2%7D%7D-%5Cfrac%7B1%7D%7B%5Csqrt%5B3%5D%7Bx%7D%7D)
3)
![\bf \begin{cases} f(x)=\sqrt{x}-5x\implies &f(x)x^{\frac{1}{2}}-5x\\\\ g(x)=5x^2-2x+\sqrt[5]{x}\implies &g(x)=5x^2-2x+x^{\frac{1}{5}} \end{cases} \\\\\\ \textit{let's multiply the terms from f(x) by each term in g(x)} \\\\\\ x^{\frac{1}{2}}(5x^2-2x+x^{\frac{1}{5}})\implies x^{\frac{1}{2}}5x^2-x^{\frac{1}{2}}2x+x^{\frac{1}{2}}x^{\frac{1}{5}}](https://tex.z-dn.net/?f=%5Cbf%20%5Cbegin%7Bcases%7D%0Af%28x%29%3D%5Csqrt%7Bx%7D-5x%5Cimplies%20%26f%28x%29x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D-5x%5C%5C%5C%5C%0Ag%28x%29%3D5x%5E2-2x%2B%5Csqrt%5B5%5D%7Bx%7D%5Cimplies%20%26g%28x%29%3D5x%5E2-2x%2Bx%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%0A%5Cend%7Bcases%7D%0A%5C%5C%5C%5C%5C%5C%0A%5Ctextit%7Blet%27s%20multiply%20the%20terms%20from%20f%28x%29%20by%20each%20term%20in%20g%28x%29%7D%0A%5C%5C%5C%5C%5C%5C%0Ax%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%285x%5E2-2x%2Bx%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%29%5Cimplies%20x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D5x%5E2-x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D2x%2Bx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7Dx%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D)
![\bf 5\sqrt{x^5}-2\sqrt{x^3}+\sqrt[10]{x^7}-25x^3+10x^2-5\sqrt[5]{x^6}](https://tex.z-dn.net/?f=%5Cbf%205%5Csqrt%7Bx%5E5%7D-2%5Csqrt%7Bx%5E3%7D%2B%5Csqrt%5B10%5D%7Bx%5E7%7D-25x%5E3%2B10x%5E2-5%5Csqrt%5B5%5D%7Bx%5E6%7D)
a)
b)
3)
795.800.913.789
1 answer:
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Please refer to my image where it shows my work as I’m explaining.
Okay, for system 1:
1. I am using the elimination method to solve. So I check if all the terms are lined up and if any are the same. I found that 2X are common in both equations.
2. The goal is to “eliminate” the term hence the name. So I can choose to add or subtract. I chose subtraction because 2 - 2 equals 0 which is our goal. Solve for the rest of the terms. This will lead to getting y =4. Refer to image for the work.
3. Last step to to find the X value. We do this by picking any of the given equations,then substitute y with 4 and solve to eventually get x = 10. Refer to image for the work.
FOR SYSTEM 2:
1. Again, I am using the elimination method to solve. I noticed that NONE of the terms are in common so I will have to intervene. You can chose any term to create a match with but I chose Y since it was the one I could use the smallest number to multiply with. When multiplying, DONT just multiply Y, multiply ALL the terms in the equation or else everything will crash.
2. Now that I have terms in common I can choose to add or subtract. I chose subtraction because 2-2 equals zero which is what we want. Solve look at image for my process which lead to X = -8
3. Last step is to find the value of Y. Chose any of the given equations in system 2 then substitute x with -8. Refer to image to see process. It lead to y = 20
To check the validity of the answers, substitute the x and y values into both equations both side of the equal side should have the same number. Hope that helped!
Okay, for system 1:
1. I am using the elimination method to solve. So I check if all the terms are lined up and if any are the same. I found that 2X are common in both equations.
2. The goal is to “eliminate” the term hence the name. So I can choose to add or subtract. I chose subtraction because 2 - 2 equals 0 which is our goal. Solve for the rest of the terms. This will lead to getting y =4. Refer to image for the work.
3. Last step to to find the X value. We do this by picking any of the given equations,then substitute y with 4 and solve to eventually get x = 10. Refer to image for the work.
FOR SYSTEM 2:
1. Again, I am using the elimination method to solve. I noticed that NONE of the terms are in common so I will have to intervene. You can chose any term to create a match with but I chose Y since it was the one I could use the smallest number to multiply with. When multiplying, DONT just multiply Y, multiply ALL the terms in the equation or else everything will crash.
2. Now that I have terms in common I can choose to add or subtract. I chose subtraction because 2-2 equals zero which is what we want. Solve look at image for my process which lead to X = -8
3. Last step is to find the value of Y. Chose any of the given equations in system 2 then substitute x with -8. Refer to image to see process. It lead to y = 20
To check the validity of the answers, substitute the x and y values into both equations both side of the equal side should have the same number. Hope that helped!
