Question

Rakesh has solved some problem of exponents in the following way: \[{{\left( {{x}^{2}} \right)}^{3}}={{x}^{{{2}^{3}}}}={{x}^{8}}\].
Tick on the correct option for the given solution.
(a)True
(b)False

Answer 1

Hint: Expand the given expression \[{{\left( {{x}^{2}} \right)}^{3}}\]on the basis of exponent power rule. Compare our answer to the answer of Rakesh and find if it's true or false.

Complete step-by-step answer:
The given solution is actually wrong.
The exponent “power rule” can be used to solve this expression. According to exponent product rule, when multiplying 2 powers that have the same base, we can add the exponents. Let us consider an example,
\[{{x}^{m}}.{{x}^{n}}={{x}^{m+n}}\]
e.g: - \[{{4}^{2}}{{.4}^{3}}={{4}^{2+3}}={{4}^{5}}\]
Similarly, the exponent “power rule” tells us that to raise a power, just multiply the exponents. For example,
\[{{\left( {{x}^{a}} \right)}^{b}}={{x}^{a.b}}={{x}^{ab}}\].
Similarly, \[{{\left( {{5}^{2}} \right)}^{3}}={{5}^{2\times 3}}={{5}^{6}}\].
We have been asked to find if Rakesh has solved the exponents in problems correctly. We have been given \[{{\left( {{x}^{2}} \right)}^{3}}\].
By the exponent “power rule” we can find the answer that,
\[{{\left( {{x}^{2}} \right)}^{3}}={{x}^{2\times 3}}={{x}^{6}}\].
The given method of solving problems i.e. how Rakesh solved is wrong.
Thus the correct answer is \[{{\left( {{x}^{2}} \right)}^{3}}={{x}^{6}}\].
\[\therefore \]Option (b) is correct.
Note: Similarly, for exponents there is quotient rule, zero rule and negative rule.
By quotient rule, \[{{x}^{m}}\div {{x}^{n}}=\dfrac{{{x}^{m}}}{{{x}^{n}}}={{x}^{m-n}}\]and \[x\ne 0\].
By zero rule, \[{{x}^{0}}=1\]and \[x\ne 0\].
By negative exponents, \[{{4}^{-2}}=\dfrac{1}{{{4}^{2}}}=\dfrac{1}{16}\].