Question

Question :-
In the above plz don't spam . Only for Brainlians ​

Answer 1

Question

$\sqrt[3]{333+\sqrt[3]{987+\sqrt[3]{2197} } }$ is equal to

(a) 21

(b) 18

(c) 7

(d) 3

__________________________

Answer

$\sqrt[3]{333+\sqrt[3]{987+\sqrt[3]{2197} } }$

To solve this, we must find the cube roots of the required number. Let's find the cube root of 2197 first.

First, we'll Prime factorize 2197

$\begin{array}{r | l} 13 & 2197 \\ \cline{2-2} 13 & 169 \\ \cline{2-2} & 13 \\ \end{array}$

2197 = 13 × 13 × 13

2197 = (13 × 13 × 13)

∛2197 = 13

∴ ∛2197 is 13.

Since we found the cube root of 2197, we'll replace it with the expression we are solving and simplify it.

⇒ $\sqrt[3]{333+\sqrt[3]{987+13 } }$

⇒ $\sqrt[3]{333+\sqrt[3]{1000 } }$

Now we'll find the cube root 1000.

We'll Prime factorize 1000 first to find the cube root.

$\begin{array}{r | l} 2 & 1000 \\ \cline{2-2} 2 & 500 \\ \cline{2-2} 2 & 250 \\ \cline{2-2} 5 & 125 \\ \cline{2-2} 5 & 25 \\ \cline{2-2} & 5\\ \end{array}$

1000 = 2 × 2 × 2 × 5 × 5 × 5

1000 = (2 × 2 × 2) × (5 × 5 × 5)

∛1000 = 2 × 5

∛1000 = 10

∴ ∛1000 is 10.

Since we found the cube root of 1000, we'll replace it with the expression we are solving and simplify it.

⇒ $\sqrt[3]{333+\sqrt[3]{1000 } }$

⇒ $\sqrt[3]{333+10 }$

⇒ $\sqrt[3]{343}$

Now we have to find the cube root of 343 in order to complete the simplification of the expression.

First, we'll Prime Factorize 343.

$\begin{array}{r | l} 7 & 343 \\ \cline{2-2} 7 & 49 \\ \cline{2-2} & 7 \\ \end{array}$

343 = 7 × 7 × 7

343 = (7 × 7 × 7)

∛343 = 7

∴ ∛343 is 7.

Since we found the cube root of 343, we'll replace it with the expression we are solving and simplify it.

⇒ $\sqrt[3]{343}$

⇒ $7$

$\bf \therefore \sqrt[3]{333+\sqrt[3]{987+\sqrt[3]{2197} } } = 7}$

∴ The answer is (c) 7.

__________________________

Answer 2

${\blue{ \sqrt[3]{333} + \sqrt[3]{987} + \sqrt[3]{2197}} =\purple{ \sqrt[3]{333 + \sqrt[3]{987 + 13} }}} \\ \\ \green{ \sqrt[3]{333 + \sqrt[3]{1000} }} =\red{ \sqrt[3]{333 + 10}} \\ \\ \orange{ \sqrt[3]{343} = 7}$

So option C is correct

$\red{REFER \: \: THE \: \: ATTACHMENT}$

FOR DETAIL ANSWER