Question

What is the exponential form of 64?

Answer 1

Hint: Here in this question, we have to write the given number in exponential form. For this, by tables of multiplication choose any number and multiply the same number by itself until to get the value 64 then write the number in the form of exponent by using laws of indices and tables of multiplication to get the required solution.

Complete step by step answer:
The exponential number is defined as the number of times the number is multiplied by itself. It is represented as $${a^n}$$, where a is the numeral and n represents the number of times the number is multiplied.
Consider the given question:
We need to write 64 in the exponential form.
Factors of 64 are: 1, 2, 4, 8, 16, 32 and 64.
Now, try to write 64 by using a table of multiplication.
$$ \Rightarrow \,\,64 = 8 \times 8$$
By using a law of indices $${a^m} \times {a^n} = {a^{m + n}}$$, then
$$ \Rightarrow \,\,\,\,64 = {8^{1 + 1}}$$
$$ \Rightarrow \,\,\,\,64 = {8^{1 + 1}}$$
$$\therefore \,\,64 = {8^2}$$
Or
$$ \Rightarrow \,\,64 = 4 \times 4 \times 4$$
Again, by law of indices
$$ \Rightarrow \,\,64 = {4^{1 + 1 + 1}}$$
$$\therefore \,\,64 = {4^3}$$
Or
$$ \Rightarrow \,\,64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2$$
Again, by law of indices
$$ \Rightarrow \,\,64 = {2^{1 + 1 + 1 + 1 + 1 + 1}}$$
$$\therefore \,\,64 = {2^6}$$
Or
$$ \Rightarrow \,\,64 = 64 \times 1$$
Again, by law of indices
$$ \Rightarrow \,\,64 = {64^1} + {64^0}$$
$$ \Rightarrow \,\,64 = {64^{1 + 0}}$$
$$\therefore \,\,64 = {64^1}$$
Hence, you can write 64 in exponential form as any of the following:
$${2^6}$$, $${4^3}$$, $${8^2}$$ and $${64^1}$$.

Note:
We have the following properties of exponents and logarithms to simplify these kinds of problems.
1. $${a^m} \times {a^n} = {a^{m + n}}$$
2. $a^{-1}=\dfrac{1}{a}$
3. $\log\left(\dfrac{a}{b}\right)= \log a - \log b$