Question

Evaluate: $ {(25)^{\dfrac{1}{3}}} \times {(5)^{\dfrac{1}{3}}} $ .

Answer 1

Hint: Here we are given two terms of the power and exponent with different bases. First of all we will convert both the terms with the same bases and then will use the additive law for the power and exponent and simplify for the resultant required value.

Complete step-by-step answer:
Take the given expression: $ {(25)^{\dfrac{1}{3}}} \times {(5)^{\dfrac{1}{3}}} $
Now, using the prime factorization convert the given composite number in the form of prime numbers.
Place, $ 25 = {5^2} $
Where square is the number multiplied with itself twice.
 $ = {({5^2})^{\dfrac{1}{3}}} \times {(5)^{\dfrac{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}} $ .
 $ = {(5)^{\dfrac{2}{3}}} \times {(5)^{\dfrac{1}{3}}} $
Now, apply the property for the laws of power and exponent in the above equation and simplify. When bases are same and in multiplicative then the powers are added, using $ {x^a} \times {x^b} = {x^{a + b}} $
 $
   = {(5)^{\dfrac{{2 + 1}}{3}}} \\
   = {(5)^{\dfrac{3}{3}}} \;
  $
Common factors from the numerator and the denominator cancels each other.
 $
   = {(5)^1} \\
   = 5 \;
  $
Hence, the required resultant solution is $ {(25)^{\dfrac{1}{3}}} \times {(5)^{\dfrac{1}{3}}} = 5 $
So, the correct answer is “5”.

Note: 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