Question

How do you write “b to the eighth power” in exponential form?

Answer 1

Hint: In this problem, we have to write “b to the eighth power” in exponential form. For this we should know how to write exponents and exponential notations. We can write the given statement in exponential notation using the exponential rules by the following way.

Complete step by step solution:
We should know that the exponential function \[x\to {{e}^{x}}\] has fundamental properties involving products,
\[\forall x,y\in \mathbb{R},{{e}^{x+y}}={{e}^{x}}\times {{e}^{y}}\]
So, from this we can see that exponential functions transforms sums into products as,
\[{{e}^{{{x}_{1}}+{{x}_{2}}+...+{{x}_{n}}}}={{e}^{{{x}_{1}}}}\times {{e}^{{{x}_{2}}}}\times {{e}^{{{x}_{3}}}}...\times {{e}^{n}}\]
We can now write the product of power property.
The product of power states that when multiplying two powers with the same base, adds the exponent.
For example,
\[{{a}^{x+y}}={{a}^{x}}\times {{a}^{y}}\]
We know that the given statement to be written in terms of exponent is,
“b to the eighth power”
We know that, eight power means ‘eight times b’
i.e.\[\Rightarrow b\times b\times b\times b\times b\times b\times b\times b\]
we can now see that, each ‘b’ has 1 raised as the power,
we can write it as,
\[\Rightarrow {{b}^{1}}\times {{b}^{1}}\times {{b}^{1}}\times {{b}^{1}}\times {{b}^{1}}\times {{b}^{1}}\times {{b}^{1}}\times {{b}^{1}}\]
We know that when multiplying two powers with the same base, adds the exponent.
\[\Rightarrow {{b}^{1}}\times {{b}^{1}}\times {{b}^{1}}\times {{b}^{1}}\times {{b}^{1}}\times {{b}^{1}}\times {{b}^{1}}\times {{b}^{1}}={{b}^{1+1+1+1+1+1+1+1}}\]
We can now ad the power terms, we get
\[\Rightarrow {{b}^{8}}\]
Therefore, “b to the eighth power” in exponential form is \[{{b}^{8}}\].

Note: Students make mistakes while writing the power, by adding them. We should always remember that when multiplying two powers with the same base, adds the exponent. We should also remember some exponential rules like product rules to solve these types of problems.