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Write(27a^-9)^-2/3wiein simplest form.

User HelloSam
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We are asked to write in its simplest form the expression :


(27a^(-9))^{(-(2)/(3))}

So we start by writing the most external power (-2/3) in root form. Recall that a denominator in the exponent's fraction is the index of the radical that has to be used. on the other hand, the numerator of the exponents fraction is a power. Also recall that when there is a NEGATIVE sign in the exponents such implies that the algebraic expression to which it is applied flips to its RECIPROCAL.

Then starting with the negative in the a^-9 we get:


((27)/(a^9))^{(-(2)/(3))}

and now using the negative sign in the external exponents, we get the reciprocal of the fractional expression as shown below:


((27)/(a^9))^{(-(2)/(3))}=((a^9)/(27))^{((2)/(3))}

Now we apply the radical form we discussed before:


((a^9)/(27))^{((2)/(3))}=(\sqrt[3]{((a^9)/(27)})^{})^2

Now we use the fact that a^9 and 27 are perfect cubes, in order to cancel the cubic root:


\sqrt[3]{((a^9)/(27)})=\frac{\sqrt[3]{a^(3\cdot3)}}{\sqrt[3]{3^3}}=(a^3)/(3)

and finally, raise this expression to the power 2 :


((a^3)/(3))^2=(a^6)/(9)

Therefore a^6/9 is the simplest form of the algebraic expression that was given

User Pat Notz
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