How do you simplify $3{a^2}{b^{ - 4}}$ and write it using only positive exponents?

Hint: In this question, we are given two exponential functions in multiplication with each other and we have to simplify this expression, that is, we have to write it easily and understandably. So, we can solve the question by using the law of exponents which states that when an exponent is raised to a negative power then we reciprocate the base keeping the power the same but dropping the negative sign. This way we can convert the only negative exponent in the given equation to positive and thus write it using only positive exponents.

Complete step-by-step solution:
We are given that $3{a^2}{b^{ - 4}}$
We know that ${a^{ - n}} = \dfrac{1}{{{a^n}}}$
So, we get –
$3{a^2}{b^{ - 4}} = \dfrac{{3{a^2}}}{{{b^4}}}$
Since 2 and 4 are positive, the obtained expression contains only positive exponents.
Hence the simplified form of $3{a^2}{b^{ - 4}}$ is $\dfrac{{3{a^2}}}{{{b^4}}}$ .

Note:We know that when the exponent is negative then we write the number as the reciprocal of the given exponent to convert it into a positive exponent, that is, ${a^{ - x}} = \dfrac{1}{{{a^x}}}$ , so ${b^{ - 4}}$ is written as $\dfrac{1}{{{b^4}}}$ . Now the students may feel the urge to apply the law of exponent $\dfrac{{{a^x}}}{{{a^y}}} = {a^{x - y}}$ but note that the base of the exponent function in the numerator is $a$ and the base of the exponent function in the denominator is $b$ . The law of exponents can be applied when the base of the two exponential functions is the same, but clearly, the two exponential functions in the obtained expression do not have the same base, so they cannot be simplified further.