Question

How do you use laws of exponents to simplify ${{\left( 3{{a}^{4}} \right)}^{3}}?$

Answer 1

To write large numbers in shorter form. So that it becomes very convenient to read, understand and compare when we use exponents. There are different laws for solving exponent but in this question we will use law of exponent power of a power i.e.
$\Rightarrow$ ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}={{a}^{mn}}$
Where $m$ and $n$ are whole numbers. If $a$ is a non-zero rational number and $m$ and $n$ are positive integers then.
$\Rightarrow$ ${{\left( {{\left( \dfrac{a}{b} \right)}^{m}} \right)}^{n}}={{\left( \dfrac{a}{b} \right)}^{mn}}$
We can use laws of exponents multiplying powers with the same exponents.
$\Rightarrow$ ${{a}^{m}}\times {{b}^{m}}={{\left( ab \right)}^{m}}$

Complete step-by-step answer:
To write large numbers in shorter form. So, it becomes very convenient to read, understand and compare when we use exponents.
We have to simplify this ${{\left( 3{{a}^{4}} \right)}^{3}}$ exponent for simplifying this exponent we will use exponent laws.
$\Rightarrow$ ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}={{a}^{m.n}}$
This is the power of a power exponent rule.
$\Rightarrow$ ${{\left( 3{{a}^{4}} \right)}^{3}}={{3}^{3}}{{\left( {{a}^{4}} \right)}^{3}}$
This simplified multiplying power with the same exponents law.
$\Rightarrow$ ${{\left( a{{b}^{m}} \right)}}={{a}^{m}}\times {{b}^{m}}$
Now,
$\Rightarrow$ ${{\left( 3{{a}^{4}} \right)}^{3}}={{3}^{3}}{{\left( {{a}^{4}} \right)}^{3}}$
$\Rightarrow$ $=27.{{a}^{4\times 3}}...$ power of power law.
$\Rightarrow$ ${{\left( 3{{a}^{4}} \right)}^{3}}=27.{{a}^{12}}$

So, after simplifying ${{\left( 3{{a}^{4}} \right)}^{3}}$ with the help of different laws of exponent we get $27{{a}^{2}}$.

Additional Information:
The exponent of a number says how many times to use a base number in a multiplication. When we deal with numbers we usually just simplify we rather deal with $27$ than with ${{3}^{3}}$ but with variables with need the exponent because we’d rather deal with ${{x}^{6}}$ with $\text{xxxxxx}$
A negative exponent means divide because the opposite of multiplying is dividing. A fractional exponent like $\dfrac{1}{n}$ means to take the ${{n}^{th}}$ root.
$\Rightarrow$ $x\dfrac{1}{n}=\sqrt[n]{x}$
There are seven exponent rules or laws of exponent that the students need to learn. Each rule shows how to solve different types of math equations. How to add, subtract, multiply and divide exponents.

Note:
Make sure you go through each exponent's rule thoroughly in class as each plays an important rule in showing exponent based questions. Simplify the equation carefully. Write the exponent rule according to the question.