Express as a mixed number:
\(\sqrt{0.64}-\sqrt[3]{-0.008}+\sqrt[4]{\frac{4}{20.25}}+\sqrt[5]{16\div\frac{1}{2}}\)
First, let's find out what each of those expressions boils down to.
square root 0.64 -- well, this is plus or minus 0.8 but I'm going to use only the positive value = 0.8 = 8/10
cube root –0.008 -- this equals –0.2 = –2/10
fourth root 4/20.25 -- I had to use the calculator to determine this is 0.666666••• = 2/3
fifth root (16 ÷ 1/2) -- same as fifth root of 32 which is 2
put them together 8/10 – (–2/10) + 2/3 + 2/1
a common denominator of 30 will work, so 24/30 + 6/30 + 20/30 + 60/30 = 110/30 = 3 20/30 = 3 2/3
It's too late for me to edit my answer above. I woke up this morning with a better reduction of the fourth root element of the equation.
the fourth root of (4/20.25)
a fourth root of a number is simply the square root of the square root of the number
we recognize that both the numerator and denominator are squares, so let's take those square roots
the square root of 4 is 2 and the square root of 20.25 is 4.5 (more about this below)
so the fourth root of (4/20.25) is the same as the square root of (2/4.5)
let's multiply (2/4.5) by (2/2) under the radical, and we have the square root of 4/9 which = 2/3
20.25 is recognizable as the square of 4.5, because 20 = 4x5
We can square any number that ends in 5 by addressing the 5 separately from the number ahead of it
Consider 4.5 as a 4 and a .5. Square the .5 to get .25 and set it aside mentally.
Take the 4 and add 1 to it and multiply the two numbers - i.e., 4x5 - to get 20.
Put the 20 and the .25 together, and get 20.25
Do this again with another example, any number that ends with 5 .... so let's go with 85
Square the 5 to get 25 and set it aside mentally
Add 1 to the 8 and multiply 8 by that 9 - i.e., 8x9 - to get 72
Combine the 72 and the 25 giving us the answer ... 85 squared is 7225
+118609 Here is my take :)
\(\sqrt{0.64}-\sqrt[3]{-0.008}+\sqrt[4]{\frac{4}{20.25}}+\sqrt[5]{16\div\frac{1}{2}}\\ =0.8-\sqrt[3]{-1*8*0.001}+\sqrt[4]{\frac{4}{2025\div100}}+\sqrt[5]{32}\\ =0.8-(-0.2)+\sqrt[4]{\frac{4*100}{2025}}+2\\ =0.8+0.2+\sqrt[2]{\frac{2*10}{45}}+2\\ =3+\sqrt[2]{\frac{2*10}{5*9}}\\ =3+\sqrt[2]{\frac{2*2}{9}}\\ =3+\frac{2}{3}\\ =3\frac{2}{3}\\\)
