Exponents and Powers — NCERT Solutions, AI Tutor & Practice

We use the rule $a^{-n} = \dfrac{1}{a^n}$.

(i) $3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}$

(ii) $(-4)^{-3} = \dfrac{1}{(-4)^3} = \dfrac{1}{-64} = -\dfrac{1}{64}$

(iii) $\left(\dfrac{1}{2}\right)^{-4} = \left(\dfrac{2}{1}\right)^{4} = 2^4 = 16$

Using the law $a^m \times a^n = a^{m+n}$:

$2^{-3} \times 2^{-2} = 2^{-3+(-2)} = 2^{-5}$

If a positive exponent is needed: $2^{-5} = \dfrac{1}{2^5} = \dfrac{1}{32}$.

Combine the left side using $a^m \times a^n = a^{m+n}$:

$\left(\frac{2}{3}\right)^{-3+5} = \left(\frac{2}{3}\right)^{2}$

So $\left(\dfrac{2}{3}\right)^{2} = \left(\dfrac{2}{3}\right)^{2m+1}$.

Since the bases are equal, the exponents are equal: $2m+1 = 2 \implies 2m = 1 \implies m = \frac{1}{2}$

First evaluate each term:

$\left(\dfrac{1}{3}\right)^{-2} = 3^2 = 9$

$\left(\dfrac{1}{2}\right)^{-3} = 2^3 = 8$

$\left(\dfrac{1}{4}\right)^{-2} = 4^2 = 16$

Now substitute: $\{9 - 8\} \div 16 = 1 \div 16 = \frac{1}{16}$

Combine the numerator: $5^{-4} \times 5^{2} = 5^{-4+2} = 5^{-2}$.

Now divide using $\dfrac{a^m}{a^n} = a^{m-n}$:

$\frac{5^{-2}}{5^{-3}} = 5^{-2-(-3)} = 5^{-2+3} = 5^{1} = 5$

Standard form is $k \times 10^n$ where $1 \le k < 10$.

(i) $0.00000564$: move the decimal 6 places to the right, so the exponent is $-6$. $0.00000564 = 5.64 \times 10^{-6}$

(ii) $52300000$: move the decimal 7 places to the left, so the exponent is $+7$. $52300000 = 5.23 \times 10^{7}$

(i) $3.02 \times 10^{-4}$: a negative exponent means move the decimal 4 places to the left. $3.02 \times 10^{-4} = 0.000302$

(ii) $4.5 \times 10^{6}$: a positive exponent means move the decimal 6 places to the right. $4.5 \times 10^{6} = 4500000$

Write both in standard form:

Paper $= 0.0016 = 1.6 \times 10^{-3}$ cm

Hair $= 0.005 = 5 \times 10^{-3}$ cm

Since $5 \times 10^{-3} > 1.6 \times 10^{-3}$, the hair is thicker.

Ratio: $\frac{5 \times 10^{-3}}{1.6 \times 10^{-3}} = \frac{5}{1.6} = 3.125$

So the hair is about $3.125$ times thicker than the paper.

Flip the reciprocals so both have the same base $\dfrac{5}{3}$:

$\left(\dfrac{3}{5}\right)^{-2} = \left(\dfrac{5}{3}\right)^{2}$

So: $\left(\frac{5}{3}\right)^{2} \times \left(\frac{5}{3}\right)^{-3} = \left(\frac{5}{3}\right)^{2+(-3)} = \left(\frac{5}{3}\right)^{-1} = \frac{3}{5}$

Combine the left side: $\left(\frac{-2}{3}\right)^{(x+1)+5} = \left(\frac{-2}{3}\right)^{9}$

Equal bases give equal exponents: $x + 1 + 5 = 9$ $x + 6 = 9 \implies x = 3$

Numerator: $(-2)^{-3} \times (-2)^{6} = (-2)^{-3+6} = (-2)^{3}$.

Divide: $\frac{(-2)^{3}}{(-2)^{2}} = (-2)^{3-2} = (-2)^{1} = -2$

So the value is $-2$.

To add, make the powers of 10 equal. Convert the electron mass to a power of $10^{-27}$:

$9.11 \times 10^{-31} = 0.000911 \times 10^{-27}$

Now add: $0.000911 \times 10^{-27} + 1.67 \times 10^{-27} = (0.000911 + 1.67) \times 10^{-27}$ $= 1.670911 \times 10^{-27} \text{ kg}$

Approximately $1.67 \times 10^{-27}$ kg, since the electron's mass is negligibly small in comparison.