Question

How do you simplify $\left( 9{{w}^{2}}{{x}^{8}} \right)\left( {{w}^{6}}{{x}^{4}} \right)$ ?

Answer 1

Hint: We have been given exponential functions consisting of variable-x and variable-w having constant positive powers. We have to simplify the given function which has two x-terms and twow-terms getting multiplied. In order to do this, we shall use certain exponential properties. Thus, we shall add the powers of the like terms to completely simplify this function.

Complete step by step solution:
Let us suppose a constant, ‘a’ has to be multiplied by itself ‘b’ times. Then we can write it in the exponential form as ${{a}^{b}}$ instead of writing $a\times a\times a\times a\times ......$upto ‘b’ times.
Exponents have their own set of rules and properties according to which they can be manipulated. One of them is that if like exponential terms are being multiplied, then their respective powers are added.
That is, ${{x}^{a}}.{{x}^{b}}={{x}^{a+b}}$ , where a and b are the powers of the base x in the exponential functions.
We have been given the exponent as a function of variable-x and variable-w, $\left( 9{{w}^{2}}{{x}^{8}} \right)\left( {{w}^{6}}{{x}^{4}} \right)$. Therefore, adding the powers of these terms, we get
$\Rightarrow \left( 9{{w}^{2}}{{x}^{8}} \right)\left( {{w}^{6}}{{x}^{4}} \right)=9\left( {{w}^{2+6}} \right)\left( {{x}^{8+4}} \right)$
$\Rightarrow \left( 9{{w}^{2}}{{x}^{8}} \right)\left( {{w}^{6}}{{x}^{4}} \right)=9\left( {{w}^{8}} \right)\left( {{x}^{12}} \right)$
$\Rightarrow \left( 9{{w}^{2}}{{x}^{8}} \right)\left( {{w}^{6}}{{x}^{4}} \right)=9{{w}^{8}}{{x}^{12}}$
Since, $3\times 3=9$ or ${{\left( 3 \right)}^{2}}=9$,
 $\Rightarrow \left( 9{{w}^{2}}{{x}^{8}} \right)\left( {{w}^{6}}{{x}^{4}} \right)={{3}^{2}}{{w}^{8}}{{x}^{12}}$
$\Rightarrow \left( 9{{w}^{2}}{{x}^{8}} \right)\left( {{w}^{6}}{{x}^{4}} \right)={{\left( 3{{w}^{4}}{{x}^{6}} \right)}^{2}}$

Therefore, $\left( 9{{w}^{2}}{{x}^{8}} \right)\left( {{w}^{6}}{{x}^{4}} \right)$ can be simplified as ${{\left( 3{{w}^{4}}{{x}^{6}} \right)}^{2}}$.

Note: One of the reasons we use multiplication is because it acts as a shorthand for successive addition. Similarly, exponents act as a shorthand for successive multiplication. Thus, on observing that $3{{w}^{4}}{{x}^{6}}\times 3{{w}^{4}}{{x}^{6}}=9{{w}^{8}}{{x}^{12}}$, we squared the term expression on the left hand side and raised it to a power of 2, that is ${{\left( 3{{w}^{4}}{{x}^{6}} \right)}^{2}}=9{{w}^{8}}{{x}^{12}}$. Thus, we can make such modifications to make the mathematical expressions look more readable.