Question

The value of $ {(256)^{0.16}} \times {(256)^{0.09}} $ is
A. $ 64 $
B. $ 256.25 $
C. $ 16 $
D. $ 4 $

Answer 1

Hint: Take the given expression and simplify using the different laws of power and exponents such as product rule and addition rule. Also the factors of the given base in the exponent and then simplify for the resultant value.

Complete step-by-step answer:
Take the given expression –
 $ {(256)^{0.16}} \times {(256)^{0.09}} $
We can observe that the above expression is in the form of – $ {x^m} \times {x^n} $ and can be simplified by using the identity $ {x^m} \times {x^n} = {x^{m + n}} $ which states that when bases are equal and there is multiplication sign in between then the powers are added.
 $ = {(256)^{0.16 + 0.09}} $
Simplify the above power and exponent-
 $ = {(256)^{0.25}} $
Convert the above decimal point in the form of the fraction –
 \[ = {(256)^{\dfrac{{25}}{{100}}}}\]
Simplify the fraction in the above expression –
 \[ = {(256)^{\dfrac{{25}}{{25 \times 4}}}}\]
Common factors from the numerator and the denominator cancel each other.
 \[ = {(256)^{\dfrac{1}{4}}}\]
Now, find the factors of the above number. We know that it is the square of the number $ 16,\;{\text{1}}{{\text{6}}^2} = 256 $
 \[ = {(16 \times 16)^{\dfrac{1}{4}}}\]
We also know that $ 16 $ is the square of $ 4,{\text{ 16 = }}{{\text{4}}^2} = 4 \times 4 $ . Substitute in the above equation –
 \[ = {(4 \times 4 \times 4 \times 4)^{\dfrac{1}{4}}}\]
The number of times the number multiplied, it gives that power to the original number. Means \[4 \times 4 \times 4 \times 4\] can be written as $ {4^4} $ . Place the values in the above equation.
 \[ = {({4^4})^{\dfrac{1}{4}}}\]
By 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}} $
 \[ = {(4)^{\dfrac{4}{4}}}\]
Common factors from the numerator and the denominator cancel each other.
 \[
   = {(4)^1} \\
   = 4 \;
 \]
So, the correct answer is “Option D”.

Note: Remember the seven basic rules of the exponent or the laws of exponents to solve these types of questions. Make sure to go through the below mentioned rules, it describes how to solve different types of exponents problems and how to add, subtract, multiply and divide the exponents.
A.Product of powers rule
B.Quotient of powers rule
C.Power of a power rule
D.Power of a product rule
E.Power of a quotient rule
F.Zero power rule
G.Negative exponent rule