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The given expression is

[tex]8^{1/2} +16^{1/4}-12^{1/2}+81^{1/4}[/tex]

We can write 8 as cube of 2, 16 ad fourth power of 2, 12 as square of 2 times 3 and 81 as fourth power of 3, that is

[tex](2^3)^{1/2} +(2^4)^{1/2} - (2^2 *3)^{1/2} + (3^4)^{1/2}[/tex]

In case of two powers, we have to multiply them.

[tex]=2^{3/2} + 2^{4/2} -2^{4/2} * 3^{1/2} + 3^{4/2}[/tex]

Simplifying the exponent, will give

[tex]=2*2^{1/2} +4 -4*3^{1/2} +9[/tex]

Combining like terms,

[tex]=13+2*2^{1/2} -4*3^{1/2}[/tex]

Answer:

[tex]2\sqrt{2} +5-2\sqrt{3}[/tex]

Step-by-step explanation:

Property of Radical expression is:

[tex](a^x)^{\frac{y}{z} } = a^{\frac{xy}{z} }[/tex]

Step to step solution of given expression is:

[tex]8^{\frac{1}{2} }+16^{\frac{1}{4} }-12^{\frac{1}{2} }+81^{\frac{1}{4} }\\\Rightarrow (2^{3})^{\frac{1}{2} }+(2^{4})^{\frac{1}{4} }-(2^{2}\times3)^{\frac{1}{2} }+(3^{4})^{\frac{1}{4} }\\\Rightarrow 2^{\frac{3}{2} } + 2 - 2\sqrt{3} +3\\\Rightarrow 2\sqrt{2} +5-2\sqrt{3}[/tex]

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