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mathdad
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Verify that both sides of the radical equation agree without using a calculator. See picture. How can this be done algebraically?
View attachment 7968
View attachment 7968
MarkFL said:I would first observe that $1+\sqrt{5}$ is a root of:
\(\displaystyle x^2-2x-4=0\)
And so, the coefficients of the expansion:
\(\displaystyle (1+\sqrt{5})^n\)
Can be found recursively via:
\(\displaystyle A_{n}=2A_{n-1}+4A_{n-2}\)
For the rational term, we have:
\(\displaystyle A_0=1,\,A_1=1\)
Hence:
\(\displaystyle A_2=2(1)+4(1)=6\)
\(\displaystyle A_3=2(6)+4(1)=16\)
\(\displaystyle A_4=2(16)+4(6)=56\)
\(\displaystyle A_5=2(56)+4(16)=176\)
And for the irrational term, we have:
\(\displaystyle A_0=0,\,A_1=1\)
\(\displaystyle A_2=2(1)+4(0)=2\)
\(\displaystyle A_3=2(2)+4(1)=8\)
\(\displaystyle A_4=2(8)+4(2)=24\)
\(\displaystyle A_5=2(24)+4(8)=80\)
And so we may conclude:
\(\displaystyle (1+\sqrt{5})^5=176+80\sqrt{5}\)
And the result follows. :)
Well, you'd have quite a few terms to manipulate; as example:RTCNTC said:What if I decided to raise both sides to the 5th power? Can it be done this way as well?
Wilmer said:Well, you'd have quite a few terms to manipulate; as example:
(a + b)^5 = a^5 + 5 a^4 b + 10 a^3 b^2 + 10 a^2 b^3 + 5 a b^4 + b^5
Wilmer said:Didn't "introduce" anything...
YOU asked about raising to 5th power...
Gave you an example.
HOKAY?!
Will do; pleasure is all mine. All yours Mark...RTCNTC said:2. I would like for you to stop commenting in my posts. To you everything is a joke.
RTCNTC said:What if I decided to raise both sides to the 5th power? Can it be done this way as well?
A radical equation is an equation that contains a radical expression, such as a square root, cube root, or other root. These equations often involve finding the value of the variable that satisfies the equation.
To solve a radical equation, first isolate the radical expression on one side of the equation. Then, raise both sides of the equation to the same power to eliminate the radical. Finally, solve for the variable by simplifying and solving the resulting equation. It is important to check your solution to ensure it is valid.
Verifying a radical equation is important in order to check the accuracy of your solution. It involves substituting the found solution back into the original equation to ensure that it satisfies the equation. If the solution does not satisfy the equation, then it is not a valid solution.
Yes, a radical equation can have more than one solution. This is because when we raise both sides of the equation to the same power, we may introduce extraneous solutions. These are solutions that may satisfy the resulting equation, but not the original equation. It is important to check all solutions to ensure they are valid.
There are some strategies that can make solving radical equations easier. For example, if the equation contains a radical with an even index, you can square both sides of the equation to eliminate the radical. Additionally, you can use the power rule to simplify expressions with exponents. However, it is important to always check your solutions to ensure they are valid.