How do you simplify ${{2}^{4}}\cdot {{2}^{5}}$? \[\]

Hint: We recall base and exponent. We recall the law exponent involving product with the same base which states that if we multiply two numbers in exponent form with same base then the product is base raised to the exponent equal to sum of exponents of the two number which means ${{a}^{m}}\cdot {{a}^{n}}={{a}^{m+n}}$. We use this law to simplify the given expression. \[\]

Complete step by step answer:
We know that if we multiply a number with itself then we can express the number in exponent form. The number is called base and the number of times we multiply with it is called exponent. If we multiply the base $b$ with itself $n$ number of times then we express the product as
\[b\times b\times b....\left( n\text{ times} \right)={{b}^{n}}\]
Here $n$ is called exponent, power or index of the base. We also know different laws for arithmetic operation among numbers in exponent form. One of them is law of product with the same base. If we multiply two numbers in exponent form with the same base then the product is raised to the exponent equal to the sum of exponents of the two numbers. If $a$ is a base then ${{a}^{m}},{{a}^{n}}$ are two numbers in exponent form with the same base. The sum of the exponents is $m+n$. So by law of product with same base we have
\[{{a}^{m}}\cdot {{a}^{n}}={{a}^{m+n}}\]
We are given in the question the numerical expression as
\[{{2}^{4}}\cdot {{2}^{5}}\]
We see that there are two numbers in exponent form which are ${{2}^{4}},{{2}^{5}}$ with the same base 2 and they are multiplied. We use law of product with same base for $a=2,m=4,n=5$ to have the simplified value as
\[{{2}^{4}}\cdot {{2}^{5}}={{2}^{4+5}}={{2}^{9}}=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=512\]

Note:
We should remember other laws of exponent like power raised to a power ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$, the law of quotient with same base $\left( \dfrac{{{a}^{m}}}{{{a}^{n}}} \right)={{a}^{m-n}}$, product raised to a power ${{\left( ab \right)}^{m}}={{a}^{m}}\cdot {{b}^{m}}$, quotient raised to a power ${{\left( \dfrac{a}{b} \right)}^{m}}=\dfrac{{{a}^{m}}}{{{b}^{m}}}$ etc. We must be careful that base and exponent cannot be zero at the same time which means ${{0}^{0}}$ is not defined.