Question

1) Find the value of x if ${{3}^{x-7}}\times {{4}^{x-4}}=768$.
2) Evaluate ${{\left( 1004 \right)}^{3}}$

Answer 1

Hint: 1) We solve this problem by first factoring 768 in terms of 3 and 4. Then we write them in powers of 3 and 4. Then we equate the powers of 3 on both sides and powers of 4 on both sides, if both the values are the same then that is the value of x.
2) We solve this problem by writing 1004 as 1000+4. Then we use the formula ${{\left( a+b \right)}^{3}}={{a}^{3}}+3{{a}^{2}}b+3a{{b}^{2}}+{{b}^{3}}$ to expand it and calculate it to find the required value.

Complete step by step answer:
1) We are given that ${{3}^{x-7}}\times {{4}^{x-4}}=768$.
First let us factorize 768.
$768=3\times 256$
Now let us consider 256.
$\begin{align}
  & \Rightarrow 256=16\times 16 \\
 & \Rightarrow 256=4\times 4\times 4\times 4 \\
\end{align}$
So, substituting the factorization of 256 in the factorization of 768, we get,
$\begin{align}
  & \Rightarrow 768=3\times 4\times 4\times 4\times 4 \\
 & \Rightarrow 768=3\times {{4}^{4}} \\
\end{align}$
So, now let us substitute this value in the given expression.
$\Rightarrow {{3}^{x-7}}\times {{4}^{x-4}}={{3}^{1}}\times {{4}^{4}}$
So, let us compare the powers of 3 and 4. Then by comparing them we get,
$\begin{align}
  & \Rightarrow x-7=1 \\
 & \Rightarrow x-4=4 \\
\end{align}$
Solving them we get,
$\begin{align}
  & \Rightarrow x=8 \\
 & \Rightarrow x=8 \\
\end{align}$
So, the value of x in both the cases is 8.
Hence the answer is 8.
2) We need to find the value of ${{\left( 1004 \right)}^{3}}$.
We can write 1004 as 1000+4.
So, ${{\left( 1004 \right)}^{3}}={{\left( 1000+4 \right)}^{3}}$
Now, let us consider the formula for ${{\left( a+b \right)}^{3}}$
${{\left( a+b \right)}^{3}}={{a}^{3}}+3{{a}^{2}}b+3a{{b}^{2}}+{{b}^{3}}$
Using this formula, we can write ${{\left( 1000+4 \right)}^{3}}$ as
$\Rightarrow {{\left( 1000+4 \right)}^{3}}={{\left( 1000 \right)}^{3}}+3{{\left( 1000 \right)}^{2}}\left( 4 \right)+3\left( 1000 \right){{\left( 4 \right)}^{2}}+{{\left( 4 \right)}^{3}}$
Solving it we get,
$\begin{align}
  & \Rightarrow {{\left( 1000+4 \right)}^{3}}=1000000000+12000000+48000+64 \\
 & \Rightarrow {{\left( 1000+4 \right)}^{3}}=1012048064 \\
\end{align}$
Hence, we get the value of ${{\left( 1004 \right)}^{3}}$ as 1012048064.
Hence the answer is 1012048064.

Note:
1) We can also write the prime factorisation for 768 as
$768=3\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=3\times {{2}^{8}}$
Then we write ${{4}^{x-4}}$ in ${{3}^{x-7}}\times {{4}^{x-4}}$ as ${{\left( {{2}^{2}} \right)}^{x-4}}={{2}^{2x-8}}$. Then we get
${{3}^{x-7}}\times {{4}^{x-4}}={{3}^{x-7}}\times {{2}^{2x-8}}$
So, equating them we get,
${{3}^{x-7}}\times {{2}^{2x-8}}=3\times {{2}^{8}}$
So, let us compare the powers of 3 and 2. Then by comparing them we get,
$\begin{align}
  & \Rightarrow x-7=1 \\
 & \Rightarrow 2x-8=8 \\
\end{align}$
Solving them we get,
$\begin{align}
  & \Rightarrow x=8 \\
 & \Rightarrow x=8 \\
\end{align}$
Hence the answer is 8.
2) One can also solve this problem by multiplying 1004 three times, that is we multiply 1004 with 1004 and multiply the obtained value with 1004 again to get the required answer. But it is a very long procedure for solving the problem.