Question

How do you use the law of exponents to simplify the expression $x{{\left( 3{{x}^{2}} \right)}^{3}}$ ?

Answer 1

Hint: The given expression can be simplified using the law of exponents. The major which should be used is the product with the same base. It basically states that when we multiply the same bases together, then we can add their exponents and keep the base the same. But in the first step, we shall use the power to a power law of exponents.

Complete step-by-step solution:
The given expression is, $x{{\left( 3{{x}^{2}} \right)}^{3}}$
To simplify this expression, we use the law of exponents.
The first law will be the product to a power.
When we raise a power to a product, we distribute the power to both factors.
${{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}$
Upon applying this law to out given expression we get,
$\Rightarrow x\left( {{3}^{3}}{{\left( {{x}^{2}} \right)}^{3}} \right)$
Now evaluate the powers further,
$\Rightarrow 27x\times {{\left( {{x}^{2}} \right)}^{3}}$
The second law we will be using is the power of a power law of exponents.
It basically states that if we have a power raised to a base and another power raised to that power, then we can directly raise the base to the product of both the powers.
It can be well explained by this,
${{\left( {{y}^{n}} \right)}^{m}}={{y}^{n\times m}}$
Upon applying this law to out given expression we get,
$\Rightarrow 27x\times \left( {{x}^{2\times 3}} \right)$
Now evaluate the powers further,
$\Rightarrow 27x\left( {{x}^{6}} \right)$
Now we use the product with the same base law of exponents.
It states that when we multiply the same bases together, then we can add their exponents keeping the base the same.
The formula is given as,
${{x}^{n}}\times {{x}^{m}}={{x}^{n\times m}}$
Upon applying this law to out given expression we get,
$\Rightarrow 27{{x}^{1+6}}$
Now evaluate the powers further,
$\Rightarrow 27{{x}^{7}}$
Hence the expression, $x{{\left( 3{{x}^{2}} \right)}^{3}}$ on simplification gives $27{{x}^{7}}$

Note: One should be familiar with all the laws of exponents to solve these types of problems and also these laws are very useful in solving logarithmic expressions too. There is another law of exponents such as dividing same bases, we subtract the exponents' etcetera.