Answer
397.5k+ views
Hint: In order to determine the simplest form of the above question ,first take the reciprocal to remove negative sign from the exponent and then write $16 = {4^2}\,and\,25 = {5^2}$ and divide the exponents using identity ${\left( {{a^m}} \right)^{\dfrac{1}{n}}} = {a^{\dfrac{m}{n}}}$ to get your required simplification.
Complete step by step solution:
We are given a rational number ${\left( {\dfrac{{25}}{{16}}} \right)^{ - \dfrac{3}{2}}}$.
In order to simplify the above rational number we will be using some of the exponential identities ,to rewrite the rational number and simplify it.
Since, any number raised to negative power , is actually the reciprocal of it .
Taking the reciprocal of our given function to remove the negative sign from the exponent value .
\[
= {\left( {\dfrac{{25}}{{16}}} \right)^{ - \dfrac{3}{2}}} \\
= {\left( {\dfrac{{16}}{{25}}} \right)^{\dfrac{3}{2}}} \\
\]
Using exponential identity which state ${a^{\dfrac{m}{n}}} = {\left( {{a^m}}
\right)^{\dfrac{1}{n}}}$ by consider m as 3 and n as 2, we get
\[ = {\left( {{{\left( {\dfrac{{16}}{{25}}} \right)}^3}} \right)^{\dfrac{1}{2}}}\]
Power get split into numerator and denominator if a power exist for some fraction
\[ = {\left( {\dfrac{{{{16}^3}}}{{{{25}^3}}}} \right)^{\dfrac{1}{2}}}\]
Since,\[{16^3} = {4^6}\]and \[{25^3} = {5^6}\]
\[
= {\left( {\dfrac{{{4^6}}}{{{5^6}}}} \right)^{\dfrac{1}{2}}} \\
= \dfrac{{{4^{\dfrac{6}{2}}}}}{{{5^{\dfrac{6}{2}}}}} \\
= \dfrac{{{4^3}}}{{{5^3}}} \\
= \dfrac{{64}}{{125}} \\
\]
Therefore, ${\left( {\dfrac{{25}}{{16}}} \right)^{ - \dfrac{3}{2}}}$in the simplest form is equal to
\[\dfrac{{64}}{{125}}\]
Additional Information:
What is a Rational Number?
A Rational number is a number which can be expressed in the form of $\dfrac{p}{q}$, where p and q are any integer value and q is not equal to 0. Such a number is known as a rational number.
Note:
1. If you want to Increase a Number by y %:
Example: On the off chance that a number is expanded by\[10{\text{ }}\% \], at that point it becomes 1.1 times of itself.
On the off chance that a number is expanded by 30%, at that point it becomes 1.3 times of itself.
2. If you want to Decrease a Number by y %:
Example: On the off chance that a number is diminished by 10%, at that point it becomes 0.90 times of itself.
On the off chance that a number is diminished by 30%, at that point it becomes 0.70 times of itself.
3. Value of the constant ”e” exponent is equal to 2.71828.
4. Don’t forget to cross check your result.
Complete step by step solution:
We are given a rational number ${\left( {\dfrac{{25}}{{16}}} \right)^{ - \dfrac{3}{2}}}$.
In order to simplify the above rational number we will be using some of the exponential identities ,to rewrite the rational number and simplify it.
Since, any number raised to negative power , is actually the reciprocal of it .
Taking the reciprocal of our given function to remove the negative sign from the exponent value .
\[
= {\left( {\dfrac{{25}}{{16}}} \right)^{ - \dfrac{3}{2}}} \\
= {\left( {\dfrac{{16}}{{25}}} \right)^{\dfrac{3}{2}}} \\
\]
Using exponential identity which state ${a^{\dfrac{m}{n}}} = {\left( {{a^m}}
\right)^{\dfrac{1}{n}}}$ by consider m as 3 and n as 2, we get
\[ = {\left( {{{\left( {\dfrac{{16}}{{25}}} \right)}^3}} \right)^{\dfrac{1}{2}}}\]
Power get split into numerator and denominator if a power exist for some fraction
\[ = {\left( {\dfrac{{{{16}^3}}}{{{{25}^3}}}} \right)^{\dfrac{1}{2}}}\]
Since,\[{16^3} = {4^6}\]and \[{25^3} = {5^6}\]
\[
= {\left( {\dfrac{{{4^6}}}{{{5^6}}}} \right)^{\dfrac{1}{2}}} \\
= \dfrac{{{4^{\dfrac{6}{2}}}}}{{{5^{\dfrac{6}{2}}}}} \\
= \dfrac{{{4^3}}}{{{5^3}}} \\
= \dfrac{{64}}{{125}} \\
\]
Therefore, ${\left( {\dfrac{{25}}{{16}}} \right)^{ - \dfrac{3}{2}}}$in the simplest form is equal to
\[\dfrac{{64}}{{125}}\]
Additional Information:
What is a Rational Number?
A Rational number is a number which can be expressed in the form of $\dfrac{p}{q}$, where p and q are any integer value and q is not equal to 0. Such a number is known as a rational number.
Note:
1. If you want to Increase a Number by y %:
Example: On the off chance that a number is expanded by\[10{\text{ }}\% \], at that point it becomes 1.1 times of itself.
On the off chance that a number is expanded by 30%, at that point it becomes 1.3 times of itself.
2. If you want to Decrease a Number by y %:
Example: On the off chance that a number is diminished by 10%, at that point it becomes 0.90 times of itself.
On the off chance that a number is diminished by 30%, at that point it becomes 0.70 times of itself.
3. Value of the constant ”e” exponent is equal to 2.71828.
4. Don’t forget to cross check your result.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science 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)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 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)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![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)
How do you graph the function fx 4x class 9 maths 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)