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# How do you simplify ${\left( {\dfrac{7}{4}} \right)^3}$?

Last updated date: 12th Sep 2024
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Hint:To simplify this question , we need to solve it step by step . starting from the parentheses with exponent over it , this means that the rational number $\dfrac{7}{4}$ ( the number which can be expressed as in the form of $\dfrac{p}{q}$ , where p is numerator and q is denominator also q$\ne$0 . ) is having cube over the numerator and the denominator also . We should first write the cube of 7 in the numerator and then the cube of 3 in the denominator . Then simplify it to get the desired answer .

Complete Step by step Solution :
The rational number with whole exponent as cube can be expressed as the number with no
exponent by writing their respective cube that is for any fraction ${\left( {\dfrac{p}{q}} \right)^n} = \dfrac{{{p^n}}}{{{q^n}}}$, p is numerator and q is denominator and n is the exponent which is distributed over the numerator and denominator .
${\left( {\dfrac{7}{4}} \right)^3} = \dfrac{7}{4} \cdot \dfrac{7}{4} \cdot \dfrac{7}{4} = \dfrac{{7 \cdot 7 \cdot 7}}{{4 \cdot 4 \cdot 4}} = \dfrac{{{7^3}}}{{{4^3}}}$ .
So , calculating the cube of the respective numbers= ${\left( {\dfrac{7}{4}} \right)^3}$=
$\dfrac{{{7^3}}}{{{4^3}}}$= $\dfrac{{343}}{{64}}$
Now , In order to solve the fraction into its simplified form =>
To simplify the fraction we need to find the Greatest Common Divisor of numerator and
denominator of the fraction $\dfrac{{343}}{{64}}$ .
The Greatest Common Divisor of 343 and 64 is 1 . Then divide the numerator and denominator by the Greatest Common Divisor .
$\dfrac{{343 \div 1}}{{64 \div 1}}$=$\dfrac{{343}}{{64}}$ .
Therefore , the reduced fraction is $\dfrac{{343}}{{64}}$. It is already in its simplest form . It can be written as $5.359375$in the decimal form .

Note:
1. Rational number is a number which is expressed in the form of $\dfrac{p}{q}$where $q \ne 0$.
2.Do the calculation properly .
3.Make sure to write the fraction into its simplest form at the end of the result.