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### Theory:

In the previous class, we have learned about the laws of exponent.

Let's have a quick recall of the laws of exponent.

The laws are:
1. Product law
According to the product law, the exponents can be added when multiplying two powers with the same base.

${a}^{n}×{a}^{m}={a}^{n+m}$, where $$a ≠ 0$$ and $$a$$, $$m$$, $$n$$ are integers.
Example:
$$10 ^2 × 10 ^5$$

Here, the base $$10$$ is same for powers. So we can add the exponents using the product law.

${10}^{2}×{10}^{5}={10}^{2+5}={10}^{7}$
2. Quotient law
The quotient law states that we can divide two powers with the same base by subtracting the exponents.

${a}^{n}:{a}^{m}=\frac{{a}^{n}}{{a}^{m}}={a}^{n-m},\phantom{\rule{0.147em}{0ex}}n>m,\phantom{\rule{0.147em}{0ex}}a\ne 0;$, where, $$a$$, $$m$$, $$n$$ are integers.
Example:
$\frac{{4}^{15}}{{4}^{13}}={4}^{15-13}={4}^{2}$

$\frac{{10}^{8}}{{10}^{5}}={10}^{8-5}={10}^{3}$
3. Power law
The power law states that when a number is raised to a power of another power, we need to multiply the powers or exponents.

${\left({a}^{n}\right)}^{m}={a}^{n×m}$, where $$a ≠ 0$$ and $$a$$, $$m$$, $$n$$ are integers.
Example:
$\begin{array}{l}{\left({4}^{5}\right)}^{3}={4}^{5×3}={4}^{15}\\ \\ {\left({10}^{4}\right)}^{2}={10}^{4+2}={10}^{6}\end{array}$
Powers with Negative Exponent
A number with negative exponent is equal to the reciprocal of the number with positive exponent.

That is, ${a}^{-n}=\frac{1}{{a}^{n}}$, here $$n$$ is an integer.
• If the negative number $$(-1)$$ raised to the negative odd power $$($$$$-1$$$$^\text{odd power}$$$$)$$, then the resultant value is negative $$(-1)$$.
• If the negative number $$(-1)$$ raised to the negative even power $$($$$$-1$$$$^\text{even power}$$$$)$$, then the resultant value is positive $$(1)$$.
Example:
${5}^{-2}=\frac{1}{{5}^{2}}=\frac{1}{5×5}=\frac{1}{25}$

${-10}^{-3}=-\frac{1}{{10}^{3}}=-\frac{1}{10×10×10}=\frac{1}{1000}$