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Express as a mixed number:

 

\(\sqrt{0.64}-\sqrt[3]{-0.008}+\sqrt[4]{\frac{4}{20.25}}+\sqrt[5]{16\div\frac{1}{2}}\)

 Feb 23, 2019
 #1
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+1

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

 Feb 23, 2019
edited by Guest  Feb 23, 2019
edited by Guest  Feb 23, 2019
 #2
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+1

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

 Feb 23, 2019
edited by Guest  Feb 23, 2019
 #3
avatar+100819 
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Here is my take :)

 

 

\(\sqrt{0.64}-\sqrt[3]{-0.008}+\sqrt[4]{\frac{4}{20.25}}+\sqrt[5]{16\div\frac{1}{2}}\\ =0.8-\sqrt[3]{-1*8*0.001}+\sqrt[4]{\frac{4}{2025\div100}}+\sqrt[5]{32}\\ =0.8-(-0.2)+\sqrt[4]{\frac{4*100}{2025}}+2\\ =0.8+0.2+\sqrt[2]{\frac{2*10}{45}}+2\\ =3+\sqrt[2]{\frac{2*10}{5*9}}\\ =3+\sqrt[2]{\frac{2*2}{9}}\\ =3+\frac{2}{3}\\ =3\frac{2}{3}\\\)

.
 Feb 23, 2019

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