Answer
Verified
376.5k+ views
Hint: To solve this question we need to know the concept of multiplication of algebraic numbers. We will be multiplying all the fractions and will be finding the numerator and denominator of the fraction. All the constant terms which are $a,b,c$ having certain powers which are given in the question will be added as the terms are being multiplied.
Complete step by step solution:
The question ask us to find the value of $x$, if the product of the three terms \[\left( \dfrac{7}{9}a{{b}^{2}} \right)\],\[\left( \dfrac{15}{7}a{{c}^{2}}b \right)\] and \[\left( -\dfrac{3}{5}{{a}^{2}}c \right)\] which is given to us is $\dfrac{-1}{x}{{a}^{4}}{{b}^{3}}{{c}^{3}}$. We are asked to find the value of $x$ by equating the product to the fraction $\dfrac{-1}{x}{{a}^{4}}{{b}^{3}}{{c}^{3}}$. The first step to solve it this problem will be to calculate the product of the terms \[\left( \dfrac{7}{9}a{{b}^{2}} \right)\],\[\left( \dfrac{15}{7}a{{c}^{2}}b \right)\] and \[\left( -\dfrac{3}{5}{{a}^{2}}c \right)\]. On calculating it we get:
$\Rightarrow \left( \dfrac{7}{9}a{{b}^{2}} \right)\times \left( \dfrac{15}{7}a{{c}^{2}}b \right)\times \left( -\dfrac{3}{5}{{a}^{2}}c \right)$
We will have fractional terms having numbers, and since each constant term which is being multiplied has a certain power, the powers on the constant number will be added. On solving this we get:
$\Rightarrow \dfrac{7}{9}\times \dfrac{15}{7}\times \left( -\dfrac{3}{5} \right)\times {{a}^{1+1+2}}\times {{b}^{2+1+0}}\times {{c}^{0+2+1}}$
$\Rightarrow \dfrac{7\times 15\times \left( -3 \right)}{9\times 7\times 5}\times {{a}^{1+1+2}}\times {{b}^{2+1+0}}\times {{c}^{0+2+1}}$
Now we will be cancelling the common terms in the numerator and the denominator, on doing this we get:
$\Rightarrow \dfrac{-1}{1}\times {{a}^{4}}\times {{b}^{3}}\times {{c}^{3}}$
So the product we get is:
$\Rightarrow \dfrac{-1}{1}{{a}^{4}}{{b}^{3}}{{c}^{3}}$
The second step is to equate the product to the term $\dfrac{-1}{x}{{a}^{4}}{{b}^{3}}{{c}^{3}}$. Now equating the two we get:
$\Rightarrow \dfrac{-1}{1}{{a}^{4}}{{b}^{3}}{{c}^{3}}=\dfrac{-1}{x}{{a}^{4}}{{b}^{3}}{{c}^{3}}$
On analysing the two, the value of $x$ we get is $1$.
$\therefore $ If the product of \[\left( \dfrac{7}{9}a{{b}^{2}} \right)\times \left( \dfrac{15}{7}a{{c}^{2}}b \right)\times \left( -\dfrac{3}{5}{{a}^{2}}c \right)\] is $\dfrac{-1}{x}{{a}^{4}}{{b}^{3}}{{c}^{3}}$ , then the value of $x$ is $1$.
Note: When a constant number having different powers are multiplied then the powers are added having the common base. For example if the three terms being multiplied are ${{a}^{m}},{{a}^{n}}$ and ${{a}^{p}}$ then the product result into ${{a}^{m+n+p}}$. On writing it mathematically we get:
$\Rightarrow {{a}^{m}}\times {{a}^{n}}\times {{a}^{n}}={{a}^{m+n+p}}$
Complete step by step solution:
The question ask us to find the value of $x$, if the product of the three terms \[\left( \dfrac{7}{9}a{{b}^{2}} \right)\],\[\left( \dfrac{15}{7}a{{c}^{2}}b \right)\] and \[\left( -\dfrac{3}{5}{{a}^{2}}c \right)\] which is given to us is $\dfrac{-1}{x}{{a}^{4}}{{b}^{3}}{{c}^{3}}$. We are asked to find the value of $x$ by equating the product to the fraction $\dfrac{-1}{x}{{a}^{4}}{{b}^{3}}{{c}^{3}}$. The first step to solve it this problem will be to calculate the product of the terms \[\left( \dfrac{7}{9}a{{b}^{2}} \right)\],\[\left( \dfrac{15}{7}a{{c}^{2}}b \right)\] and \[\left( -\dfrac{3}{5}{{a}^{2}}c \right)\]. On calculating it we get:
$\Rightarrow \left( \dfrac{7}{9}a{{b}^{2}} \right)\times \left( \dfrac{15}{7}a{{c}^{2}}b \right)\times \left( -\dfrac{3}{5}{{a}^{2}}c \right)$
We will have fractional terms having numbers, and since each constant term which is being multiplied has a certain power, the powers on the constant number will be added. On solving this we get:
$\Rightarrow \dfrac{7}{9}\times \dfrac{15}{7}\times \left( -\dfrac{3}{5} \right)\times {{a}^{1+1+2}}\times {{b}^{2+1+0}}\times {{c}^{0+2+1}}$
$\Rightarrow \dfrac{7\times 15\times \left( -3 \right)}{9\times 7\times 5}\times {{a}^{1+1+2}}\times {{b}^{2+1+0}}\times {{c}^{0+2+1}}$
Now we will be cancelling the common terms in the numerator and the denominator, on doing this we get:
$\Rightarrow \dfrac{-1}{1}\times {{a}^{4}}\times {{b}^{3}}\times {{c}^{3}}$
So the product we get is:
$\Rightarrow \dfrac{-1}{1}{{a}^{4}}{{b}^{3}}{{c}^{3}}$
The second step is to equate the product to the term $\dfrac{-1}{x}{{a}^{4}}{{b}^{3}}{{c}^{3}}$. Now equating the two we get:
$\Rightarrow \dfrac{-1}{1}{{a}^{4}}{{b}^{3}}{{c}^{3}}=\dfrac{-1}{x}{{a}^{4}}{{b}^{3}}{{c}^{3}}$
On analysing the two, the value of $x$ we get is $1$.
$\therefore $ If the product of \[\left( \dfrac{7}{9}a{{b}^{2}} \right)\times \left( \dfrac{15}{7}a{{c}^{2}}b \right)\times \left( -\dfrac{3}{5}{{a}^{2}}c \right)\] is $\dfrac{-1}{x}{{a}^{4}}{{b}^{3}}{{c}^{3}}$ , then the value of $x$ is $1$.
Note: When a constant number having different powers are multiplied then the powers are added having the common base. For example if the three terms being multiplied are ${{a}^{m}},{{a}^{n}}$ and ${{a}^{p}}$ then the product result into ${{a}^{m+n+p}}$. On writing it mathematically we get:
$\Rightarrow {{a}^{m}}\times {{a}^{n}}\times {{a}^{n}}={{a}^{m+n+p}}$
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE