Answer
414.6k+ views
Hint: Here in this question we need to find the value of${{\left( 256 \right)}^{0.16}}\times {{\left( 256 \right)}^{0.09}}$ . We will initially simplify the expression using simple basic exponent formulae ${{a}^{x}}.{{a}^{y}}={{a}^{x+y}}$and ${{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}$. The factors of 256 are \[1,2,4,8,16,256,128,32,64\] because
$256=1\times 256=2\times 128=4\times 64=8\times 32=16\times 16$. We will use this for further simplifications which will lead us to the answer.
Complete step by step answer:
The value of${{\left( 256 \right)}^{0.16}}\times {{\left( 256 \right)}^{0.09}}$.
As we know that from the basic concept of exponents that the product of a raised to the power x and a raised to the power y is equal to a raised to the power of sum of x and y where x and y are positive integers and a is a real number.
Since we know that${{a}^{x}}.{{a}^{y}}={{a}^{x+y}}$.
This can be written as${{\left( 256 \right)}^{0.16+0.09}}$.
As we know that 0.16+0.09=0.25 we can simply write this as${{\left( 256 \right)}^{0.25}}$.
Since $0.25=\dfrac{1}{4}$ we can write this simply as${{\left( 256 \right)}^{\dfrac{1}{4}}}$.
As we know that from the basic concept of exponents that the value of a raised to the power $\dfrac{1}{n}$ is equal to the value of a raised to the power n where n positive integer and a is a real number.
As we know that ${{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}$this can be written as${{\left( 256 \right)}^{\dfrac{1}{4}}}=\sqrt[4]{256}$.
The factors of 256 are \[1,2,4,8,16,256,128,32,64\]because
$256=1\times 256=2\times 128=4\times 64=8\times 32=16\times 16$.
We will use the grouping $256=4\times 4\times 4\times 4$and get the desired values.
Since we know that $256={{4}^{4}}$we can write this as $\sqrt[4]{256}=\sqrt[4]{{{4}^{4}}}$
Hence we can say that$\sqrt[4]{{{4}^{4}}}=4$.
So we will end up having the conclusion that the value of ${{\left( 256 \right)}^{0.16}}\times {{\left( 256 \right)}^{0.09}}$ is 4.
So, the correct answer is “Option A”.
Note: While solving these types of questions we should remember the properties of exponents. Let us discuss the different basic properties of exponents. There are 8 basic properties when a and b are real numbers and x and y and n are positive integers.
1. ${{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}$
2. ${{a}^{x}}.{{a}^{y}}={{a}^{x+y}}$
3. ${{a}^{0}}=1,\forall a\ne 0$
4. ${{a}^{x}}\div {{a}^{y}}={{a}^{x-y}}$
5. ${{\left( {{a}^{x}} \right)}^{y}}={{a}^{x\times y}}$
6. ${{\left( \dfrac{a}{b} \right)}^{n}}={{a}^{n}}\times {{b}^{-n}}$
7. ${{\left( a\times b \right)}^{n}}={{a}^{n}}\times {{b}^{n}}$
8. ${{\left( \dfrac{a}{b} \right)}^{-n}}={{\left( \dfrac{b}{a} \right)}^{n}}$
$256=1\times 256=2\times 128=4\times 64=8\times 32=16\times 16$. We will use this for further simplifications which will lead us to the answer.
Complete step by step answer:
The value of${{\left( 256 \right)}^{0.16}}\times {{\left( 256 \right)}^{0.09}}$.
As we know that from the basic concept of exponents that the product of a raised to the power x and a raised to the power y is equal to a raised to the power of sum of x and y where x and y are positive integers and a is a real number.
Since we know that${{a}^{x}}.{{a}^{y}}={{a}^{x+y}}$.
This can be written as${{\left( 256 \right)}^{0.16+0.09}}$.
As we know that 0.16+0.09=0.25 we can simply write this as${{\left( 256 \right)}^{0.25}}$.
Since $0.25=\dfrac{1}{4}$ we can write this simply as${{\left( 256 \right)}^{\dfrac{1}{4}}}$.
As we know that from the basic concept of exponents that the value of a raised to the power $\dfrac{1}{n}$ is equal to the value of a raised to the power n where n positive integer and a is a real number.
As we know that ${{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}$this can be written as${{\left( 256 \right)}^{\dfrac{1}{4}}}=\sqrt[4]{256}$.
The factors of 256 are \[1,2,4,8,16,256,128,32,64\]because
$256=1\times 256=2\times 128=4\times 64=8\times 32=16\times 16$.
We will use the grouping $256=4\times 4\times 4\times 4$and get the desired values.
Since we know that $256={{4}^{4}}$we can write this as $\sqrt[4]{256}=\sqrt[4]{{{4}^{4}}}$
Hence we can say that$\sqrt[4]{{{4}^{4}}}=4$.
So we will end up having the conclusion that the value of ${{\left( 256 \right)}^{0.16}}\times {{\left( 256 \right)}^{0.09}}$ is 4.
So, the correct answer is “Option A”.
Note: While solving these types of questions we should remember the properties of exponents. Let us discuss the different basic properties of exponents. There are 8 basic properties when a and b are real numbers and x and y and n are positive integers.
1. ${{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}$
2. ${{a}^{x}}.{{a}^{y}}={{a}^{x+y}}$
3. ${{a}^{0}}=1,\forall a\ne 0$
4. ${{a}^{x}}\div {{a}^{y}}={{a}^{x-y}}$
5. ${{\left( {{a}^{x}} \right)}^{y}}={{a}^{x\times y}}$
6. ${{\left( \dfrac{a}{b} \right)}^{n}}={{a}^{n}}\times {{b}^{-n}}$
7. ${{\left( a\times b \right)}^{n}}={{a}^{n}}\times {{b}^{n}}$
8. ${{\left( \dfrac{a}{b} \right)}^{-n}}={{\left( \dfrac{b}{a} \right)}^{n}}$
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)