Uh-Oh! That’s all you get for now. We would love to personalise your learning journey. Sign Up to explore more. Sign Up or Login Skip for now Uh-Oh! That’s all you get for now. We would love to personalise your learning journey. Sign Up to explore more. Sign Up or Login Skip for now Answer  Hint:Cube root a given number is a number when multiplied three times will give the given number. Use the prime factorization method to write the given number in form of prime factors, now take triplets of the same numbers and multiply them to get the cube root of the given number. Complete step-by-step answer: We know that the cube root of any number can be calculated using prime factorization of that number by taking pairs of 3 same numbers (prime numbers).Basically, it means we can get the given number by multiplying the cube root of that number by three times.So, we get$\text{cube root}\times \text{cube root}\times \text{cube root}=\text{Number given}$So, let us find the prime factors of 4096 by factorizing it.Prime factorization means we will use prime numbers for dividing the given numbers and hence, we know that prime number is a number which have only two factors i.e. 1 and the number itself. Smallest prime number is 2, so try to divide it by 2 initially then look for other higher prime numbers if required.So, the number given in the question is 4096.Hence, prime factorization can be given as$\begin{align} & 2\left| \!{\underline {\, 4096 \,}} \right. \\ & 2\left| \!{\underline {\, 2048 \,}} \right. \\ & 2\left| \!{\underline {\, 1024 \,}} \right. \\ & 2\left| \!{\underline {\, 512 \,}} \right. \\ & 2\left| \!{\underline {\, 256 \,}} \right. \\ & 2\left| \!{\underline {\, 128 \,}} \right. \\ & 2\left| \!{\underline {\, 64 \,}} \right. \\ & 2\left| \!{\underline {\, 32 \,}} \right. \\ & 2\left| \!{\underline {\, 16 \,}} \right. \\ & 2\left| \!{\underline {\, 8 \,}} \right. \\ & 2\left| \!{\underline {\, 4 \,}} \right. \\ & \text{ 2} \\ \end{align}$Hence, 4097 can be written in form of factors$4096=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$Now, we need to take the same factors in form of triplet means take 3 same numbers at only times and hence multiply then to get the cube root. So, we can observe that\[4096=\text{2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2 }\!\!\times\!\!\text{ 2}\]Now, cube root of 4096 can be given asCube root of $4096=2\times 2\times 2\times 2=16$So, we can verify the cube root of 4096 by multiplying 16 to three times. Hence, the cube root of 4096 is 16.Note: Don’t miss any term while writing the given number in factorizing form. And take pairs of 3 same numbers not 2. Pairs of 2 same numbers and multiplication of them will give square root not cubic root. So, be clear with the terminologies and concepts. Use only prime numbers for factorization of the given number. Using any other number except prime will give the wrong answer and lead to the wrong solution as well. So, be clear and careful with it as well. |
Show that 4096 is a perfect cube by prime factorisation method.
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