Question

item 3
write the expression as an expression with a rational exponent. if the expression is equivalent to an integer, write that integer.
\\(\left(\sqrt3{-216}\
ight)^2\\)

Explanation:

Step1: Recall the radical to exponent rule

The formula for converting a radical to a rational exponent is $\sqrt[n]{a}=a^{\frac{1}{n}}$, and when we have a power of a radical, $( \sqrt[n]{a} )^m=a^{\frac{m}{n}}$. Here, we have $(\sqrt[3]{- 216})^{2}$. First, identify $n = 3$, $a=-216$ and $m = 2$.

Step2: Apply the rule

Using the rule $( \sqrt[n]{a} )^m=a^{\frac{m}{n}}$, substitute $n = 3$, $a=-216$ and $m = 2$ into the formula. So we get $(-216)^{\frac{2}{3}}$. Now, we can also simplify the base. We know that $-216=(-6)^3$ (since $(-6)\times(-6)\times(-6)=-216$). Then $(-216)^{\frac{2}{3}}=((-6)^3)^{\frac{2}{3}}$.

Step3: Simplify the exponent

Using the power - of - a - power rule $(a^m)^n=a^{m\times n}$, we have $((-6)^3)^{\frac{2}{3}}=(-6)^{3\times\frac{2}{3}}=(-6)^2$.

Step4: Calculate the final value

Calculate $(-6)^2=36$.

Answer:

36