Cubes and Cube Roots Test - 14

On prime factorising, we get,

          $$=3^3\times 2^3=6^3$$.

Then, $$-216$$ $$=(-6)^3$$.

Therefore, cube root of $${-216}$$,

i.e. $$\sqrt[3]{-216}=\sqrt[3]{(-6)^3}= -6$$.

Therefore, option $$A$$ is correct.

On prime factorising, we get,

$${9261}=\underline{3\times 3\times 3}\times \underline{7\times 7\times 7}=3^3 \times 7^3 $$.

Then, value of $$\sqrt[3]{9261}$$ is:

$$\sqrt[3]{9261}=\sqrt[3]{3^3\times 7^3}= 3 \times 7=21$$.

Therefore, option $$B$$ is correct.

On prime factorising, we get,

$$3375=\underline{3\times 3\times 3}\times \underline {5\times5\times5}$$ $$=3^3 \times 5^3=15^3$$.

Then, $$-3375$$ $$=(-15)^3$$.

Therefore, value of $$\sqrt[3]{-3375}$$ is:

$$\sqrt[3]{-3375}=\sqrt[3]{(-15)^3}= -15$$.

Therefore, option $$B$$ is correct.

$${\textbf{Step -1: Given, number is  - 1.}}$$

               $${\text{As we know that, prime factorization of 1 is,}}$$

               $$1 = 1 \times 1 \times 1 = {1^3}$$

               $${\text{So, prime factorization of  - 1 will be,}}$$

               $$ - 1 = \left( { - 1} \right) \times \left( { - 1} \right) \times \left( { - 1} \right) = {\left( { - 1} \right)^3}$$

               $$ \Rightarrow  - 1 = {\left( { - 1} \right)^3}$$

$${\textbf{Step -2: Taking cube root on both sides}}$$

               $$ \Rightarrow \sqrt[3]{{ - 1}} = \sqrt[3]{{{{\left( { - 1} \right)}^3}}} =  - 1.$$

               $${\text{Thus, cube root of  - 1 is  - 1}}{\text{.}}$$ 

$${\textbf{ Hence, Option (B)}}{\textbf{  -1, is correct answer.}}$$

On prime factorising, we get,

          $$=3^3\times 2^3=6^3$$.

Then, $$-216$$ $$=(-6)^3$$.

Therefore, cube root of $${-216}$$ is:

$$\sqrt[3]{-216}=\sqrt[3]{(-6)^3}= -6$$.

Therefore, option $$D$$ is correct.

Prime factorising $$8575$$, we get,

$$8575= 5 \times 5 \times 7 \times 7 \times 7 $$

          $$= 5^2 \times 7 ^3$$.

We know, a perfect cube has multiples of $$3$$ as powers of prime factors.

Here, number of $$5$$'s is $$2$$ and number of $$7$$'s is $$3$$.

So we need to multiply another $$7$$ to the factorization to make $$8575$$ a perfect cube.

Hence, the smallest number by which $$8575$$ must be multiplied to obtain a perfect cube is $$5$$.

Hence, option $$C$$ is correct.

On prime factorising, we get,

       $$=2^3\times 2^3\times 2^3=8^3$$.

Then, $$-512$$ $$=(-8)^3$$.

Therefore, cube root of $${-512}$$,

i.e. $$\sqrt[3]{-512}=\sqrt[3]{(-8)^3}= -8$$.

Therefore, option $$B$$ is correct.

Given, the number is $$4276$$.

Here, the units digit is $$6$$.

We know, the cube of $$6$$, i.e. $$6^3=216$$, whose units place is $$6$$.

Therefore, the units digit of the cube of $$4276$$ is $$6$$.

Hence, option $$A$$ is correct.