Key Points

For any non-zero integer 'a' and a positive integer 'm', a negative exponent indicates the reciprocal of the base raised to the positive exponent. The formula is $a^{-m} = \frac{1}{a^m}$.

A number raised to a negative exponent, $a^{-m}$, is the multiplicative inverse of the same number raised to the positive exponent, $a^m$. Their product is $a^m \times a^{-m} = a^{m-m} = a^0 = 1$.

Any non-zero number raised to the power of zero is always equal to 1. The rule is $a^0 = 1$, where $a \neq 0$.

When multiplying two exponential terms with the same base, keep the base and add the exponents. The formula is $a^m \times a^n = a^{m+n}$.

When dividing two exponential terms with the same base, keep the base and subtract the exponents. The formula is $a^m \div a^n = a^{m-n}$.

When raising a power to another power, keep the base and multiply the exponents. The formula is $(a^m)^n = a^{mn}$.

When a product of bases is raised to a power, distribute the exponent to each base. The formula is $(a \times b)^m = a^m \times b^m$.

When a quotient is raised to a power, distribute the exponent to both the numerator and the denominator. The formula is $(\frac{a}{b})^m = \frac{a^m}{b^m}$, where $b \neq 0$.

A fraction raised to a negative exponent is equal to its reciprocal raised to the corresponding positive exponent. The rule is $(\frac{a}{b})^{-m} = (\frac{b}{a})^m$.

A number is expressed in standard form as $k \times 10^n$, where $k$ is a decimal number such that $1 \leq k < 10$, and $n$ is an integer. This form is used for very large or very small numbers.

For numbers between 0 and 1, the exponent $n$ in standard form is negative. For example, the number $0.000056$ can be written as $5.6 \times 10^{-5}$.

For numbers greater than or equal to 10, the exponent $n$ in standard form is positive. For example, the number $942,000,000$ can be written as $9.42 \times 10^8$.

To convert a number from standard form to its usual form, move the decimal point. If the exponent of 10 is positive, move the decimal to the right. If it is negative, move the decimal to the left.