Question

find each of the following using prime factorisation.
(a) (sqrt3{3375})
(b) (sqrt3{1728})
(c) (sqrt3{5832})
(d) (sqrt3{8000})

Explanation:

Part (a): $\boldsymbol{\sqrt[3]{3375}}$

Step 1: Prime Factorize 3375

We find the prime factors of 3375.
We start by dividing by 5: $3375\div5 = 675$; $675\div5 = 135$; $135\div5 = 27$. Then divide 27 by 3: $27\div3 = 9$; $9\div3 = 3$; $3\div3 = 1$.
So, $3375 = 5\times5\times5\times3\times3\times3 = 5^3\times3^3$.

Step 2: Simplify the Cube Root

Using the property of cube roots $\sqrt[3]{a^3\times b^3}=\sqrt[3]{a^3}\times\sqrt[3]{b^3}=a\times b$, we have:
$\sqrt[3]{3375}=\sqrt[3]{5^3\times3^3}=\sqrt[3]{5^3}\times\sqrt[3]{3^3}=5\times3 = 15$.

Part (b): $\boldsymbol{\sqrt[3]{1728}}$

Step 1: Prime Factorize 1728

We find the prime factors of 1728.
Divide by 2: $1728\div2 = 864$; $864\div2 = 432$; $432\div2 = 216$; $216\div2 = 108$; $108\div2 = 54$; $54\div2 = 27$. Then divide 27 by 3: $27\div3 = 9$; $9\div3 = 3$; $3\div3 = 1$.
So, $1728 = 2\times2\times2\times2\times2\times2\times3\times3\times3=2^6\times3^3=(2^2)^3\times3^3 = 4^3\times3^3$.

Step 2: Simplify the Cube Root

Using the property of cube roots $\sqrt[3]{a^3\times b^3}=\sqrt[3]{a^3}\times\sqrt[3]{b^3}=a\times b$, we have:
$\sqrt[3]{1728}=\sqrt[3]{4^3\times3^3}=\sqrt[3]{4^3}\times\sqrt[3]{3^3}=4\times3 = 12$.

Part (c): $\boldsymbol{\sqrt[3]{5832}}$

Step 1: Prime Factorize 5832

We find the prime factors of 5832.
Divide by 2: $5832\div2 = 2916$; $2916\div2 = 1458$; $1458\div2 = 729$. Then divide 729 by 3: $729\div3 = 243$; $243\div3 = 81$; $81\div3 = 27$; $27\div3 = 9$; $9\div3 = 3$; $3\div3 = 1$.
So, $5832 = 2\times2\times2\times3\times3\times3\times3\times3\times3=2^3\times(3^2)^3=2^3\times9^3$.

Step 2: Simplify the Cube Root

Using the property of cube roots $\sqrt[3]{a^3\times b^3}=\sqrt[3]{a^3}\times\sqrt[3]{b^3}=a\times b$, we have:
$\sqrt[3]{5832}=\sqrt[3]{2^3\times9^3}=\sqrt[3]{2^3}\times\sqrt[3]{9^3}=2\times9 = 18$.

Part (d): $\boldsymbol{\sqrt[3]{8000}}$

Answer:

s:
(a) $\boldsymbol{15}$
(b) $\boldsymbol{12}$
(c) $\boldsymbol{18}$
(d) $\boldsymbol{20}$