What is sqrt((9cos^2(θ))^3) ?

The solution to sqrt((9cos^2(θ))^3) is 27cos^3(θ)

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simplify sqrt((9cos^2(θ))^3)

simplify

(9 cos^{2} (θ))^{3}

27 cos^{3} (θ)

Solution steps

(9 cos^{2} (θ))^{3}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((9 cos^{2} (θ))^{3})^{\frac{1}{2}}

Apply exponent rule:

(a^{b})^{c} = a^{b c},

assuming

a \geq 0

= (9 cos^{2} (θ))^{3 \cdot \frac{1}{2}}

3 \cdot \frac{1}{2} = \frac{3}{2}

= (9 cos^{2} (θ))^{\frac{3}{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 9^{\frac{3}{2}} (cos^{2} (θ))^{\frac{3}{2}}

9^{\frac{3}{2}} = 27

= 27 (cos^{2} (θ))^{\frac{3}{2}}

(cos^{2} (θ))^{\frac{3}{2}} : cos^{3} (θ)

= 27 cos^{3} (θ)

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