Simplify`(m^3*m^-4)^-2 +(2m)^(1/2) *m^(1/4) *m^(5/4)`

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` ` Please state your workings clearly.Thanks

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To multiply powers of the same base, add exponents. To find power of a power, multiply exponents.

Therefore `(m^3 * m^-4)^-2 = (m^-1)^-2 =m^2`

Power of a product is the product of powers:

`(2m)^(1/2) = 2^(1/2) * m^(1/2) = sqrt(2) * m^(1/2)`

And

`(2m)^(1/2) * m^(1/4) * m^(5/4) = sqrt(2) * m^(1/2 + 1/4 + 5/4)= sqrt(2) * m^2`

Therefore

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= `m^2 + sqrt(2) * m^2 =(1+ sqrt(2))*m^2`

**The answer is** `(1 + sqrt(2))* m^2`

There are a couple exponent properties that need to be known for this problem:

`(x^a)(x^b)=(x^(a+b))` and

`(x^a)^b=x^(ab)`

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`((m^3)(m^(-4)))^(-2) + (2m)^(1/2)m^(1/4)m^(5/4)`

So with the properties we can simplify the first part:

`(m^(-1))^(-2) + (2m)^(1/2)m^(1/4)m^(5/4)`

Now, we can distribute the exponents on the outside of the parenthesis:

`m^2 + sqrt(2)m^(1/2)m^(1/4)m^(5/4)`

Then you can combine all the separate M's on the right side:

`m^2 + sqrt(2)m^2` which is equal to `(1+sqrt(2))*m^2`

**Thus**, `((m^3)(m^(-4)))^(-2) + (2m)^(1/2)m^(1/4)m^(5/4)=(1+sqrt2)*m^2`

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