Question

answer this with full explanation
wrong answers would be reported...​

Answer 1

Using the power rule, the number can be expressed without radicals.

This requires the radicand to be positive.

$\sf{\sqrt[4]{\sqrt[3]{3^2} } }$

$\sf{=\sqrt[4]{(3^2)^{\frac{1}{3} }}$

$\sf{=(3^2)^{\frac{1}{3} \times \frac{1}{4} }}$

$\sf{=3^\frac{1}{6} }$

So the correct answer is choice b.

More information:

Restrictions of the base on the power rule?

Do we get $\sf{-3}$ after we use the power rule to calculate $\sf{\sqrt{(-3)^2} }$?

No. The correct answer will be $\sf{3}$ because the power rule applies when the base is positive numbers only.

The correct calculation will be $\sf{\sqrt{(-3)^2} =\sqrt{9} =\sqrt{3^2} =3}$.

The power rule has different restrictions.

• Real power

These four rules are the basic power rule.

The base should be positive.

$\implies\sf{a^x\div a^y=a^{x-y}}$

$\implies\sf{(a^x)^y=a^{xy}}$

$\implies\sf{(ab)^x=a^xb^x}$

• Rational power

This is the power rule involving radicals.

The base should be positive.

$\implies\sf{a^{\frac{1}{n} }=\sqrt[n]{a} }$

$\implies\sf{a^{\frac{m}{n} }=\sqrt[n]{a^m} }$

• Integer power

The base should be real numbers. This is the difference between the rules above.

$\implies\sf{a^0=1(a\neq 0)}$

$\implies\sf{a^{-n}=\dfrac{1}{a^n} }$