Question

Which number is greater, \[{{\left( 30 \right)}^{100}}\] or \[{{\left( 2 \right)}^{567}}\] .

Answer 1

Hint: To find the greatest among \[{{\left( 30 \right)}^{100}}\] and \[{{\left( 2 \right)}^{567}}\] , let us first consider \[{{\left( 2 \right)}^{5}}=32\] . We know that \[32>\text{ }30\], hence, \[{{\left( 2 \right)}^{5}}>\text{ }30\] . Raising both the sides to the power of 100 gives \[{{\left( {{\left( 2 \right)}^{5}} \right)}^{100}}>\text{ }{{\left( 30 \right)}^{100}}\] . Using the identity ${{({{a}^{m}})}^{n}}={{a}^{mn}}$ , we will get \[{{\left( 2 \right)}^{100\ \ \times \ \ 5}}>\text{ }{{\left( 30 \right)}^{100}}\] . Obviously, \[{{\left( 2 \right)}^{567}}\] is greater than \[{{\left( 2 \right)}^{500}}\] .By comparing this to the previous set, we will get the required answer.

Complete step by step answer:
We need to find the greatest among \[{{\left( 30 \right)}^{100}}\] and \[{{\left( 2 \right)}^{567}}\] .
Since the powers are large numbers, it is good not to go for expanding each. Let us simplify the powers first.
We know that, \[{{\left( 2 \right)}^{5}}=2\times 2\times 2\times 2\times 2\] is equal to \[32\] .
And, we know that \[32>\text{ }30\]
Therefore, \[{{\left( 2 \right)}^{5}}>\text{ }30\]
Now, raising both the sides to the power of 100, we get:-
\[\Rightarrow {{\left( {{\left( 2 \right)}^{5}} \right)}^{100}}>\text{ }{{\left( 30 \right)}^{100}}\]
We know that ${{({{a}^{m}})}^{n}}={{a}^{mn}}$ .
Hence, the above equality becomes,
\[\Rightarrow {{\left( 2 \right)}^{100\ \ \times \ \ 5}}>\text{ }{{\left( 30 \right)}^{100}}\]
Multiplying, we will get
\[\Rightarrow {{\left( 2 \right)}^{500}}>\text{ }{{\left( 30 \right)}^{100}}\]
Now, let us consider \[{{\left( 2 \right)}^{567}}\] .
We know that \[{{\left( 2 \right)}^{567}}\ \ \text{must}\ \ \text{be}\ \ \text{greater}\ \ \text{than}\ \ {{\left( 2 \right)}^{500}}\]
So, \[{{2}^{567}}\ \ >\ \ {{2}^{500}}\]
Now, as \[{{\left( 2 \right)}^{5}}>\text{ }30\]
Similarly, \[{{\left( 2 \right)}^{500}}>\text{ }{{\left( 30 \right)}^{100}}\]
And, as \[{{2}^{567}}\ \ >\ \ {{2}^{500}}\]
Therefore, \[{{\left( 2 \right)}^{567}}>\text{ }{{\left( 30 \right)}^{100}}\]

Hence, \[{{\left( 2 \right)}^{567}}\] is the greater number.

Note: Students often do mistakes in solving the numbers with exponents. For example, if we have a number \[{{3}^{6}}\] . So, if we are asked to solve this, some students multiply the base by the exponent, i.e. if we are talking about the example, multiply 3 by 6. This is wrong. They need to multiply 3 by 3 six times.
\[\Rightarrow 3\ \ \times \ \ 3\ \ \times \ \ 3\ \ \times \ \ 3\ \ \times \ \ 3\ \ \times \ \ 3\ =729\]
Also the properties and rules of exponents must be thorough. In problems with higher exponents, do not expand them. Instead , use the properties to simplify them.