Question

What is $4$ to the ${{10}^{th}}$ power?

Answer 1

Hint: If a is a positive integer and b is any real number, then a to the power b is given by
${{a}^{b}}=a\times a\times a\times ....b$ times
Here, a is called the ‘base’ and b is called the ‘exponent’ or the ‘power’.
We have been asked the ${{10}^{th}}$ power of 4. Here, we have $a=4$ and $b=10$ ,so we have to multiply 4 ten times to get the required value.

Complete step-by-step answer:
Now, 4 raised to the power 10 can be expanded as:
${{4}^{10}}=4\times 4\times 4\times 4\times 4\times 4\times 4\times 4\times 4\times 4$
Taking cubes of 4 together, we have the following expanded form
$\begin{align}
  & {{4}^{10}}=\left( 4\times 4\times 4 \right)\left( 4\times 4\times 4 \right)\left( 4\times 4\times 4 \right)\left( 4 \right) \\
 & {{4}^{10}}={{4}^{3}}\times {{4}^{3}}\times {{4}^{3}}\times 4 \\
 & {{4}^{10}}=64\times 64\times 64\times 4 \\
\end{align}$
Now, beyond this, we would need to proceed through long multiplication method.
$\begin{align}
& \,\,\,\,\,\,\,\,\,\,\,\,64 \\
& \,\,\,\text{ }\underline{\times \text{64}} \\
& \,\,\,\,\,\,\,\,\,256 \\
& \underline{+384\text{X}} \\
& \,\,\,\,\,\,4096 \\
\end{align}$
After multiplying 64 with 64, we are left with multiplication of 64 and 4 to the obtained value,so multiplying 64 to 4096, we get
$\begin{align}
& \,\,\,\,\,\,\,\,\,\,\,\,\,4096 \\
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\times 64} \\
& \,\,\,\,\,\,\,\,\,\,16384 \\
& \, \underline{+24576\text{X}} \\
& \,\,\,\,\,\,\,\underline{262144} \\
\end{align}$
Now, we are left to multiply 4 with the obtained result to get the value of ${{4}^{10}}$ , so proceeding further, we get
$\begin{align}
  & \text{ }262144 \\
 & \underline{\,\,\,\,\,\,\,\,\,\, \times \text{ 4}} \\
 & 1048576 \\
\end{align}$
Hence, the value of 4 raised to the $10^{th}$ power is ${{4}^{10}}=1048576$ .

Note: Here, the long-multiplication process used is lengthy and time-consuming. Since it involves multiplying large digit numbers together, it should be done carefully without undertaking any calculation errors.
We can also write the given expression combining different powers of 4 as per our convenience so as to prevent heavy calculations.
For example, on combining cubes of 4 in two groups, and multiplying the remaining 4s together, we can get
${{4}^{10}}=\left( 4\times 4\times 4 \right)\left( 4\times 4\times 4 \right)\left( 4\times 4\times 4\times 4 \right)$
$\begin{align}
  & {{4}^{10}}={{4}^{3}}\times {{4}^{3}}\times {{4}^{4}} \\
 & {{4}^{10}}=64\times 256\times 64 \\
\end{align}$
Here, although the numbers are large in magnitude, we have reduced the number of times we need to do the long multiplication process.
Or, we could combine squares of 4 in three groups and multiply the remaining 4s together, so we can get
 ${{4}^{10}}=\left( 4\times 4 \right)\left( 4\times 4 \right)\left( 4\times 4 \right)\left( 4\times 4\times 4\times 4 \right)$
 $\begin{align}
  & {{4}^{10}}={{4}^{2}}\times {{4}^{2}}\times {{4}^{2}}\times {{4}^{4}} \\
 & {{4}^{10}}=16\times 16\times 16\times 256 \\
\end{align}$
Here, the numbers are small in magnitude and easy to multiply but the number of times we need to do the long multiplication process has increased. In short, there can be more than one way possible to multiply a certain set of numbers, it totally depends on which way you chose as per your convenience. Answers in all the cases will obviously be the same.