Help with this question

Could you help me with this question?

5(sqrt5) can be written in the form 5^k

What is the value of k

\(\begin{array}{rcll} && 5\cdot \sqrt{5} \qquad &| \qquad \sqrt{5} = 5^{\frac12} \\ &=& 5\cdot 5^{\frac12} \qquad &| \qquad 5 = 5^1 \\ &=& 5^1 \cdot 5^{\frac12} \qquad &| \qquad a^b\cdot a^c = a^{b+c} \\ &=& 5^{1+\frac12} \qquad &| \qquad 1 = \frac22\\ &=& 5^{\frac22+\frac12}\qquad &| \qquad \frac22+\frac12 = \frac{2+1}{2} \\ &=& 5^{\frac{2+1}{2}} \\ &=& 5^{\frac{3}{2}} \qquad &| \qquad \frac{3}{2} = 1,5\\ &=& 5^{1,5} \\\\ \mathbf{k}& \mathbf{=}& \mathbf{1,5} \end{array}\)

\(5^k=5\sqrt5=5^1\times5^{\frac{1}{2}}=5^{1+\frac{1}{2}}=5^{\frac{3}{2}}\)

therefore k = \(\frac{3}{2} \)

Hi Goaterino,

Welome to the Web2.0calc forum   

My answer is really just the same as Heureka's and Max's.        Thanks Heureka and Max :)

If you still do not understand it would be helpful if you try to explain which bit you do not understand :)

5(sqrt5) can be written in the form 5^k

What is the value of k

Things you must know

\(\sqrt5=5^{1/2}\\ 5=5^1\\ 5^a*5^b=5^{a+b} \)

so

\(5(sqrt5)\\ =5^1*\sqrt5\\ =5^1*5^{1/2}\\ =5^1*5^{0.5}\\ =5^{1+0.5}\\ =5^{1.5}\)

So k=1.5

Heureka has used a comma instead of a decimal point - some countries do that :)