Answer

Verified

339k+ views

**Hint:**We can observe that the given equation contains one cubic variable in subtraction. In algebra we have the formula ${{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)$. But in the given equation we have additionally coefficient for the variable and a constant, so we need to factorize the coefficient of the given variable, constant at the same time and we will convert the given equation in form of ${{a}^{3}}+{{b}^{3}}$, by applying some exponential rules, then we will use the algebraic formula and write the given equation as the product of its factors.

**Complete step-by-step solution:**

Given equation $16{{x}^{3}}-250$.

Considering the coefficient of ${{x}^{3}}$, which is $16$. Factoring $16$.

Dividing $16$ with $2$. We will get $8$ as quotient, so we can write $16$ as

$16=2\times 8$.

Now dividing $8$ with $2$. We will get $4$ as quotient, so we can write $8$ as

$8=2\times 4$.

Now dividing $4$ with $2$. We will get $2$ as quotient, so we can write $4$ as

$4=2\times 2$.

From the all the above values we can write the number $16$ as

$\begin{align}

& 16=2\times 8 \\

& \Rightarrow 16=2\times 2\times 4 \\

& \Rightarrow 16=2\times 2\times 2\times 2 \\

\end{align}$

We have an exponential rule $a\times a\times a\times ....n\text{ times}={{a}^{n}}$, then we will have

$\therefore 16={{2}^{4}}$

Considering the constant which is $250$. Factoring $250$.

Dividing $250$ with $2$. We will get $125$ as quotient, so we can write

$250=2\times 125$

We can write the value $125$ as $125=5\times 5\times 5$. From the exponential rule $a\times a\times a\times ....n\text{ times}={{a}^{n}}$, we can write $125$ as $125={{5}^{3}}$

Now the value $250$ will be written as

$250=2\times {{5}^{3}}$.

Now the given equation is modified as

$16{{x}^{3}}-250={{2}^{4}}{{x}^{3}}-2\times {{5}^{3}}$

Taking $2$ as common from the above equation, then we will have

$\Rightarrow 16{{x}^{3}}-250=2\left( {{2}^{3}}{{x}^{3}}-{{5}^{3}} \right)$

We have the exponential rule ${{a}^{m}}{{b}^{m}}={{\left( ab \right)}^{m}}$, then we will get

$\Rightarrow 16{{x}^{3}}-250=2\left[ {{\left( 2x \right)}^{3}}-{{5}^{3}} \right]$

Applying the formula ${{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)$ in the above equation, then we will get

$\begin{align}

& \Rightarrow 16{{x}^{3}}-250=2\left[ \left( 2x-5 \right)\left( {{\left( 2x \right)}^{2}}+\left( 2x \right)\left( 5 \right)+{{5}^{2}} \right) \right] \\

& \Rightarrow 16{{x}^{3}}-250=2\left( 2x-5 \right)\left( 4{{x}^{2}}+10x+25 \right) \\

\end{align}$

**Note:**When we multiply the above obtained factors, we need to get the given equation as result. If you don’t get the given equation as a result, then our solution is not correct. If they are the same then our result is correct.

Recently Updated Pages

Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Write a stanza wise summary of money madness class 11 english CBSE

Which places in India experience sunrise first and class 9 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Which neighbouring country does not share a boundary class 9 social science CBSE

What is Whales collective noun class 10 english CBSE