MHB How to Solve the Radical Equation Using Fifth Root and Square Root?

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To solve the radical equation frt{176 + 80sqrt{5}} = 1 + sqrt{5}, both sides are raised to the 5th power, resulting in 176 + 80sqrt{5} = (1 + sqrt{5})^5. The calculation of (1 + sqrt{5})^5 is broken down into steps, first finding (1 + sqrt{5})^2 = 6 + 2sqrt{5}, then squaring that result to get 56 + 24sqrt{5}, and finally multiplying by (1 + sqrt{5}) to arrive at 176 + 80sqrt{5}. An alternative method using the binomial theorem confirms that (1 + sqrt{5})^5 equals 176 + 80sqrt{5}. Thus, the equation is verified as correct.
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Let frt = fifth root

Let sqrt = square root

Show that frt{176 + 80sqrt{5}} = 1 + sqrt{5}

Do I raise both sides to the 5th power?

Can someone get me started?
 
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Yes. By raising both sides to the 5th power we get:
$$\left (\sqrt[5]{176+80\sqrt{5}}\right )^5=\left (1+\sqrt{5}\right )^5 \Rightarrow 176+80\sqrt{5}=\left (1+\sqrt{5}\right )^5$$

So, we have to calculate $\left (1+\sqrt{5}\right )^5$ and show that it is equal to $176+80\sqrt{5}$:

We have that $$\left (1+\sqrt{5}\right )^5=\left (1+\sqrt{5}\right )^2\cdot \left (1+\sqrt{5}\right )^2\cdot \left (1+\sqrt{5}\right )$$

the first term is:
$$\left (1+\sqrt{5}\right )^2=1+2\cdot \sqrt{5}+5=6+2\sqrt{5}$$

The first two terms are:
$$\left (1+\sqrt{5}\right )^2\cdot \left (1+\sqrt{5}\right )^2=\left (\left (1+\sqrt{5}\right )^2\right )^2=\left (6+2\sqrt{5}\right )^2=36+2\cdot 6\cdot 2\sqrt{5}+4\cdot 5=36+24\sqrt{5}+20 =56+24\sqrt{5}$$

The whole expression is:
\begin{align*}\left (1+\sqrt{5}\right )^2\cdot \left (1+\sqrt{5}\right )^2\cdot \left (1+\sqrt{5}\right )&=\left (\left (1+\sqrt{5}\right )^2\cdot \left (1+\sqrt{5}\right )^2\right )\cdot \left (1+\sqrt{5}\right ) \\ & =\left (56+24\sqrt{5}\right )\cdot \left (1+\sqrt{5}\right ) \\ & =56+24\sqrt{5}+56\sqrt{5}+24\cdot 5 \\ & =56+80\sqrt{5}+120 \\ & =176+80\sqrt{5}\end{align*}
 
An alternate approach would be to use the binomial theorem:

$$(1+\sqrt{5})^5=\sum_{k=0}^{5}\left({5 \choose k}\sqrt{5}^k\right)$$

$$(1+\sqrt{5})^5=1+5\sqrt{5}+50+50\sqrt{5}+125+25\sqrt{5}=176+80\sqrt{5}$$

And so:

$$\sqrt[5]{176+80\sqrt{5}}=\sqrt[5]{(1+\sqrt{5})^5}=1+\sqrt{5}$$
 
mathmari said:
Yes. By raising both sides to the 5th power we get:
$$\left (\sqrt[5]{176+80\sqrt{5}}\right )^5=\left (1+\sqrt{5}\right )^5 \Rightarrow 176+80\sqrt{5}=\left (1+\sqrt{5}\right )^5$$

So, we have to calculate $\left (1+\sqrt{5}\right )^5$ and show that it is equal to $176+80\sqrt{5}$:

We have that $$\left (1+\sqrt{5}\right )^5=\left (1+\sqrt{5}\right )^2\cdot \left (1+\sqrt{5}\right )^2\cdot \left (1+\sqrt{5}\right )$$

the first term is:
$$\left (1+\sqrt{5}\right )^2=1+2\cdot \sqrt{5}+5=6+2\sqrt{5}$$

The first two terms are:
$$\left (1+\sqrt{5}\right )^2\cdot \left (1+\sqrt{5}\right )^2=\left (\left (1+\sqrt{5}\right )^2\right )^2=\left (6+2\sqrt{5}\right )^2=36+2\cdot 6\cdot 2\sqrt{5}+4\cdot 5=36+24\sqrt{5}+20 =56+24\sqrt{5}$$

The whole expression is:
\begin{align*}\left (1+\sqrt{5}\right )^2\cdot \left (1+\sqrt{5}\right )^2\cdot \left (1+\sqrt{5}\right )&=\left (\left (1+\sqrt{5}\right )^2\cdot \left (1+\sqrt{5}\right )^2\right )\cdot \left (1+\sqrt{5}\right ) \\ & =\left (56+24\sqrt{5}\right )\cdot \left (1+\sqrt{5}\right ) \\ & =56+24\sqrt{5}+56\sqrt{5}+24\cdot 5 \\ & =56+80\sqrt{5}+120 \\ & =176+80\sqrt{5}\end{align*}

Thank you very much for providing such a detailed reply.

- - - Updated - - -

MarkFL said:
An alternate approach would be to use the binomial theorem:

$$(1+\sqrt{5})^5=\sum_{k=0}^{5}\left({5 \choose k}\sqrt{5}^k\right)$$

$$(1+\sqrt{5})^5=1+5\sqrt{5}+50+50\sqrt{5}+125+25\sqrt{5}=176+80\sqrt{5}$$

And so:

$$\sqrt[5]{176+80\sqrt{5}}=\sqrt[5]{(1+\sqrt{5})^5}=1+\sqrt{5}$$

Thanks a million. I had no idea that the binomial theorem is somehow connected here. I learn something new every time I visit this site.
 
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