Question

Evaluate : $ \sqrt[3]{{27 \times 64}} $

Answer 1

Hint: First of all find the prime factors for the terms expressed inside the cube root in the form of composite number and express the prime factors in short form using the power and exponent and then apply the power to power rule and simplify.

Complete step-by-step answer:
Take the given expression: $ \sqrt[3]{{27 \times 64}} $
Find the factors for the term,
 $ 27 = 3 \times 3 \times 3 = {3^3} $ and $ 64 = 4 \times 4 \times 4 = {4^3} $
Prime factorization is the method of finding the prime number which are multiplied together to get the original number. It can be found out by another method called factor tree method.
 $ = \sqrt[3]{{{3^3} \times {4^3}}} $
When powers are equal and bases are different then bases are multiplied with the whole power
 $ = \sqrt[3]{{{{(3 \times 4)}^3}}} $
The above expression can be re-written as –
 $ = {(3 \times 4)^{3 \times \frac{1}{3}}} $
Now, apply the property of the Power rule: to raise Power to power you have to multiply the exponents such as - $ {\left( {{x^a}} \right)^b} = {x^{ab}} $ .
 $ = {(3 \times 4)^{\frac{3}{3}}} $
Common factors from the numerator and the denominator cancel each other.
 $
   = (3 \times 4) \\
   = 12 \;
  $
Hence, the required answer is $ \sqrt[3]{{27 \times 64}} = 12 $
So, the correct answer is “12”.

Note: Be good in multiples and power and exponent. Remember the most important and basic seven rules of the exponent or the laws of exponents to solve these types of questions. Make sure to go through all the below mentioned rules which describe how to solve different types of exponents problems and how to add, subtract, multiply and divide the exponents.
I.Product of powers rule
II.Quotient of powers rule
III.Power of a power rule
IV.Power of a product rule
V.Power of a quotient rule
VI.Zero power rule
VII.Negative exponent rule