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# How do you find the product of $\left( 10x \right)\left( 4{{x}^{7}} \right)$?

Last updated date: 25th Feb 2024
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Hint: Take the product of the constant terms, i.e., 10 and 4 separately and the product of the variables x and ${{x}^{7}}$ separately. Use the formula of exponents and powers given as: - ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ to simplify the product of variables. Now, multiply the resultant of the two products considered at the initial step to get the answer.

Complete step by step answer:
Here, we have been provided with a pair of monomials: 10x and $4{{x}^{7}}$. We have been asked to find their product. But first let us understand the meaning of the term ‘monomial’.
Now, monomial is an expression that contains only one term. For example: - a, 6n, 5x, $9{{y}^{2}}$, $10{{a}^{3}}{{b}^{2}}{{c}^{5}}$ etc. These are all examples of monomials as they contain only one term. A monomial converts into a binomial when the variables are separated with a (+) or (-) sign.
Now, let us come to the question. We have two monomials 10x and $4{{x}^{7}}$. So, taking their product, we get,
$\Rightarrow \left( 10x \right)\left( 4{{x}^{7}} \right)=\left( 10\times 4 \right)\left( x\times {{x}^{7}} \right)$
Here, we have grouped the constants together and the variables together and considering their product, we get,
$\Rightarrow \left( 10x \right)\left( 4{{x}^{7}} \right)=40\left( x\times {{x}^{7}} \right)$
Using the formula: - ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$, we get,
\begin{align} & \Rightarrow \left( 10x \right)\left( 4{{x}^{7}} \right)=40{{x}^{1+7}} \\ & \Rightarrow \left( 10x \right)\left( 4{{x}^{7}} \right)=40{{x}^{8}} \\ \end{align}

Hence, the required product is $40{{x}^{8}}$.

Note: One may check the answer by substituting some small value to the variable x. For example: - let us substitute x = 1, then in the L.H.S. we will have $\left( 10\times 1 \right)\left( 4\times {{1}^{7}} \right)=10\times 4=40$. Now, in the R.H.S. we will have $4\times {{1}^{8}}=40$. Do not substitute any larger value of x because the exponent is 8 which is a large number. You must remember the formulas of exponents and powers like: - ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$, $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$, ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ etc.