Simplify the expression `2\sqrt{a^{2}b^{8}}\left(ab^{3}\right)^{-1}`You may type many lines to show your work. Enter equations inside the text using the square-root button below.

Simplify the expression 2sqrta2b8leftab3right1You may type many lines to show your work Enter equations inside the text using the squareroot button below class=

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ANSWER

2b

EXPLANATION

To simplify this expression, we have to apply some of the exponents' properties. First, the square root is a fractional exponent,

[tex]\sqrt{x}=x^{1/2}[/tex]

So we can rewrite the expression as,

[tex]2\sqrt{a^2b^8}(ab^3)^{-1}=2(a^2b^8)^{1/2}(ab^3)^{-1}[/tex]

Then, we can distribute the exponents into the multiplication,

[tex](xy)^z=x^zy^z[/tex]

In this problem,

[tex]2(a^2b^8)^{1/2}(ab^3)^{-1}=2(a^2)^{1/2}(b^8)^{1/2}(a)^{-1}(b^3)^{-1}[/tex]

Exponents of exponents are multiplied,

[tex](x^y)^z=x^{yz}[/tex]

In this problem,

[tex]2(a^2)^{1/2}(b^8)^{1/2}(a)^{-1}(b^3)^{-1}=2\cdot a^{2\cdot1/2}\operatorname{\cdot}b^{8\operatorname{\cdot}1/2}\operatorname{\cdot}a^{-1}\operatorname{\cdot}b^{3\operatorname{\cdot}(-1)}[/tex]

Simplify if possible,

[tex]2\cdot a^{2\cdot1/2}\operatorname{\cdot}b^{8\operatorname{\cdot}1/2}\operatorname{\cdot}a^{-1}\operatorname{\cdot}b^{3\operatorname{\cdot}(-1)}=2\cdot a^1\operatorname{\cdot}b^4\operatorname{\cdot}a^{-1}\operatorname{\cdot}b^{-3}[/tex]

Now, the product of two powers with the same base is equal to the base raised to the sum of the exponents,

[tex]x^y\cdot x^z=x^{y+z}[/tex]

In this problem,

[tex]2\cdot a^1\operatorname{\cdot}b^4\operatorname{\cdot}a^{-1}\operatorname{\cdot}b^{-3}=2\cdot a^{1-1}\operatorname{\cdot}b^{4-3}[/tex]

Solve the subtractions,

[tex]2\cdot a^{1-1}\operatorname{\cdot}b^{4-3}=2\cdot a^0\cdot b^1=2b[/tex]

Hence, the simplified expression is 2b.