Simplifying Radicals

Question

Hi first of all thank you very much

 

I have a question its about simplifying radicals

 

2^3 square root 128x^4 subtract 2x63 square root 54x

 

can you please 

Answer 1

Do you mean 2 ³√(128x⁴)  -  2*6 ³√(54x) ?  

It looks like your problem may really involve cube roots, not square roots.

So, this is 2 times the cube root of the whole term (128x⁴) under the radical minus 2 times the cube root of the whole term (54x)  under the radical.

If this is the case, you will want to first simplify each individual radical (cube root)

In order to simplify a cube-root radical, you will want to look for perfect cubes that are factors.  Generally, you want to find the largest perfect cube factor.  Do you know what a perfect cube is?  8 is a perfect cube because 2*2*2 = 8, or 2³ = 8.   27 is a perfect cube because 3*3*3=27, or 3³=27.  For simplifying cube-root radicals, it may be best if you know (or have memorized) some perfect cubes.  I've already mentioned 2³ and 3³.  4³=64 and 5³=125.  If you remember at least these 4 perfect cubes, that should help.

Now for the variable part of the terms, you also want to look for perfect cubes.  If the variable has an exponent that is a multiple of 3, like 3, 6, or 9, it would be a perfect cube.  So, x³, x⁶, x⁹ are all perfect cubes.

Now, let's look at simplifying the radical expression that you have.

Think:  What is the largest perfect cube that is a factor of 128?  Well, 64*2 = 128, and 64 IS a perfect cube.

Then, to deal with the variable part of the term, think about the largest perfect cube that is a factor.  We can write x⁴ as x³x, since x⁴ is the same as x*x*x*x.

So, the cube root of 128x⁴ can be rewritten as the cube root of 64*2*x³*x.  Since 4³=64, the cube root of 64 is 4.  Also, the cube root of x³ is x.

Thus, the first term 2 ³√(128x⁴) is the same as 2* ³√(64*2)(x³*x) = 2*(4x) ³√(2x) since the cube root of 64 is 4 and the cube root of x³ is x.  Then simplifying this term further gives 8x ³√(2x).

Can you simplify the second term by yourself now?

Then, you will want to combine the terms IF they are like terms (have the same variable and radical parts)

Hope this helps.