c) \( \sqrt{1500} :
d) \sqrt[4]{2592} :
e) \sqrt[3]{160} :
f) \sqrt{405} =
g) \sqrt{1200} :
h) \sqrt[3]{625} :
i) \sqrt{512} :

Q: c) \( \sqrt{1500} : d) \sqrt[4]{2592} : e) \sqrt[3]{160} : f) \sqrt{405} = g) \sqrt{1200} : h) \sqrt[3]{625} : i) \sqrt{512} :

Vamos resolver cada uma das raízes: c) $\sqrt{1500}$ Para simplificar $\sqrt{1500}$, podemos fatorar 1500: \[ 1500 = 2^2 \times 3 \times 5^3 \] Assim, $\sqrt{1500} = \sqrt{2^2 \times 3 \times 5^3} = 2 \times 5 \times \sqrt{3 \times 5} = 10\sqrt{15}$. d) $\sqrt[4]{2592}$ Fatorando 2592: \[ 2592 = 2^5 \times 3^4 \] Assim, $\sqrt[4]{2592} = \sqrt[4]{2^5 \times 3^4} = 3 \times \sqrt[4]{2^5} = 3 \times 2 \times \sqrt[4]{2} = 6\sqrt[4]{2}$. e) $\sqrt[3]{160}$ Fatorando 160: \[ 160 = 2^5 \times 5 \] Assim, $\sqrt[3]{160} = \sqrt[3]{2^5 \times 5} = 2^1 \times \sqrt[3]{2^2 \times 5} = 2 \times \sqrt[3]{20}$. f) $\sqrt{405}$ Fatorando 405: \[ 405 = 3^4 \times 5 \] Assim, $\sqrt{405} = \sqrt{3^4 \times 5} = 3^2 \times \sqrt{5} = 9\sqrt{5}$. g) $\sqrt{1200}$ Fatorando 1200: \[ 1200 = 2^4 \times 3 \times 5^2 \] Assim, $\sqrt{1200} = \sqrt{2^4 \times 3 \times 5^2} = 2^2 \times 5 \times \sqrt{3} = 20\sqrt{3}$. h) $\sqrt[3]{625}$ Fatorando 625: \[ 625 = 5^4 \] Assim, $\sqrt[3]{625} = \sqrt[3]{5^4} = 5 \times \sqrt[3]{5}$. i) $\sqrt{512}$ Fatorando 512: \[ 512 = 2^9 \] Assim, $\sqrt{512} = \sqrt{2^9} = 2^4 \times \sqrt{2} = 16\sqrt{2}$. Essas são as simplificações das raízes apresentadas.

Question

c) \( \sqrt{1500} : 
d) \sqrt[4]{2592} : 
e) \sqrt[3]{160} : 
f) \sqrt{405} = 
g) \sqrt{1200} : 
h) \sqrt[3]{625} : 
i) \sqrt{512} :

Accepted Answer

Vamos resolver cada uma das raízes:

c) $\sqrt{1500}$

Para simplificar $\sqrt{1500}$, podemos fatorar 1500:
$$ 1500 = 2^2 \times 3 \times 5^3 $$
Assim, $\sqrt{1500} = \sqrt{2^2 \times 3 \times 5^3} = 2 \times 5 \times \sqrt{3 \times 5} = 10\sqrt{15}$.

d) $\sqrt[4]{2592}$

Fatorando 2592:
$$ 2592 = 2^5 \times 3^4 $$
Assim, $\sqrt[4]{2592} = \sqrt[4]{2^5 \times 3^4} = 3 \times \sqrt[4]{2^5} = 3 \times 2 \times \sqrt[4]{2} = 6\sqrt[4]{2}$.

e) $\sqrt[3]{160}$

Fatorando 160:
$$ 160 = 2^5 \times 5 $$
Assim, $\sqrt[3]{160} = \sqrt[3]{2^5 \times 5} = 2^1 \times \sqrt[3]{2^2 \times 5} = 2 \times \sqrt[3]{20}$.

f) $\sqrt{405}$

Fatorando 405:
$$ 405 = 3^4 \times 5 $$
Assim, $\sqrt{405} = \sqrt{3^4 \times 5} = 3^2 \times \sqrt{5} = 9\sqrt{5}$.

g) $\sqrt{1200}$

Fatorando 1200:
$$ 1200 = 2^4 \times 3 \times 5^2 $$
Assim, $\sqrt{1200} = \sqrt{2^4 \times 3 \times 5^2} = 2^2 \times 5 \times \sqrt{3} = 20\sqrt{3}$.

h) $\sqrt[3]{625}$

Fatorando 625:
$$ 625 = 5^4 $$
Assim, $\sqrt[3]{625} = \sqrt[3]{5^4} = 5 \times \sqrt[3]{5}$.

i) $\sqrt{512}$

Fatorando 512:
$$ 512 = 2^9 $$
Assim, $\sqrt{512} = \sqrt{2^9} = 2^4 \times \sqrt{2} = 16\sqrt{2}$.

Essas são as simplificações das raízes apresentadas.