Exponents — Statement I-II Type Questions

Statement 1 : Exponential form of 2048 is 2¹².

Statement 2 : For any number 'a' and a positive integer 'n', we define aⁿ = a × a × a × .... a(n times).

Which of the following options is correct ?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

By prime factorisation,

$\begin{array}{l|r} 2 & 2048 \\ \hline 2 & 1024 \\ \hline 2 & 512 \\ \hline 2 & 256 \\ \hline 2 & 128 \\ \hline 2 & 64 \\ \hline 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \end{array}$

2048 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2¹¹, not 2¹².

Thus, Statement 1 is false.

For any number 'a' and a positive integer 'n', aⁿ = a × a × a × .... a (n times) is the correct definition.

Thus, Statement 2 is true.

∴ Statement 1 is false, and statement 2 is true.

Hence, Option 4 is the correct option.

Statement 1 : $\left(a^2\right)^{\dfrac{1}{4}} \div \left(a^3\right)^{\dfrac{1}{6}} = 1$

Statement 2 : If exponent of the exponent is given, then we multiply the exponents, i.e. (aⁿ)^m = a^mn.

Which of the following options is correct ?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

Solving,

$\left(a^2\right)^{\frac{1}{4}} \div \left(a^3\right)^{\frac{1}{6}}\\[1em] = a^{2 \times \frac{1}{4}} \div a^{3 \times \frac{1}{6}}\\[1em] = a^{\frac{1}{2}} \div a^{\frac{1}{2}}\\[1em] = a^{\frac{1}{2} - \frac{1}{2}}\\[1em] = a^0\\[1em] = 1$

Thus, Statement 1 is true.

The power law states that (aⁿ)^m = a^mn, which is correct.

Thus, Statement 2 is true.

∴ Both the statements are true.

Hence, Option 1 is the correct option.