Question

Express the following as a power with base 2.
(a) \[{{8}^{-5}}\]
(b) \[{{\left( 16 \right)}^{3}}\]

Answer 1

Hint: Use the prime factorization method to write prime factors of 8 in part (a) and 16 in part (b). Write the given numbers as the product of their primes. Write the product of these primes as the exponent of a single prime factor, i.e. 2. Now, use the formula of the topic ‘exponent and powers’ given as, \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\], to get the answer. Here, ‘a’ is called the base, and ‘m’ and ‘n’ are exponents.

Complete step-by-step solution
Here, we have been provided with two expressions and we have to write them as a power with base 2. So, let us consider them one – by – one.
(a) \[{{8}^{-5}}\]
Here, we have been provided with the expression \[{{8}^{-5}}\]. Here, we have base 8 and we have to convert it in base 2. So, using prime factorization to write 8 as the product of its primes, we get,
\[\Rightarrow 8=2\times 2\times 2\]
Here, 2 is multiplied 3 times. So, it can be written as: -
\[\Rightarrow 8={{2}^{3}}\] - (1)
So, substituting the value of 8 from equation (1) in the given expression, we get,
\[\begin{align}
  & \Rightarrow {{8}^{-5}}={{2}^{3\times \left( -5 \right)}} \\
 & \Rightarrow {{8}^{-5}}={{2}^{-15}} \\
\end{align}\]
Hence, it is converted as a power with base 2.
(b) \[{{\left( 16 \right)}^{3}}\]
Here, we have been provided with the expression \[{{\left( 16 \right)}^{3}}\]. So, using prime factorization to write 16 as the product of its primes, we get,
\[\Rightarrow 16=2\times 2\times 2\times 2\]
Here, 2 is multiplied 4 times. So, it can be written as: -
\[\Rightarrow 16={{2}^{4}}\] - (2)
So, substituting the value of 16 from equation (2) in the given expression, we get,
\[\Rightarrow {{\left( 16 \right)}^{3}}={{\left( {{2}^{4}} \right)}^{3}}\]
Now, applying the formula: - \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\], we get,
\[\begin{align}
  & \Rightarrow {{\left( 16 \right)}^{3}}={{2}^{4\times 3}} \\
 & \Rightarrow {{\left( 16 \right)}^{3}}={{2}^{12}} \\
\end{align}\]
Hence, it is converted as a power with base 2.

Note: One may note that it is very important to determine the prime factors of 8 and 16 to solve the above question. Note that here we have to express them as a power with base 2. So, no other prime factors like 3, 5, 7….. should come there, otherwise we will get a wrong answer. For this reason, the prime factorization method is used. We must remember some basic formulas of the topic ‘exponents and powers’ like: - \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\], \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}},\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\], as they are used everywhere.