Power PlayClass 8 Mathematics Important Points

A number in exponential form is written as $a^n$, where 'a' is the base and 'n' is the exponent or power. It represents the base 'a' multiplied by itself 'n' times, for example, $5^4 = 5 \times 5 \times 5 \times 5$.

To multiply powers with the same base, you add their exponents. The rule is $a^m \times a^n = a^{m+n}$.

To divide powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator. The rule is $\frac{a^m}{a^n} = a^{m-n}$, where $a \neq 0$.

To raise a power to another power, you multiply the exponents. The rule is $(a^m)^n = a^{m \times n}$.

To find the power of a product, you raise each factor to that power and then multiply. The rule is $(a \times b)^m = a^m \times b^m$.

To find the power of a quotient, you raise both the numerator and the denominator to that power. The rule is $(\frac{a}{b})^m = \frac{a^m}{b^m}$, where $b \neq 0$.

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

A number raised to a negative exponent is the reciprocal of the number raised to the corresponding positive exponent. The rule is $a^{-n} = \frac{1}{a^n}$, where $a \neq 0$.

The reciprocal of a number with a negative exponent is the number with a positive exponent. The rule is $\frac{1}{a^{-n}} = a^n$, where $a \neq 0$.

Scientific notation (or standard form) is used to express very large or very small numbers. A number is written as $x \times 10^y$, where $1 \leq x < 10$ and $y$ is an integer.

To write a number in scientific notation, move the decimal point to get a number between 1 and 10. The number of places the decimal moved becomes the exponent of 10. For example, $59,760,000 = 5.976 \times 10^7$.

To convert a number from scientific notation to standard form, move the decimal point to the right for a positive exponent or to the left for a negative exponent. For example, $1.496 \times 10^{11} = 149,600,000,000$.

Numbers can be written as a sum of the products of their digits and powers of 10. For example, $4756 = (4 \times 10^3) + (7 \times 10^2) + (5 \times 10^1) + (6 \times 10^0)$.

Decimal numbers use negative powers of 10 for the fractional part. For example, $561.93 = (5 \times 10^2) + (6 \times 10^1) + (1 \times 10^0) + (9 \times 10^{-1}) + (3 \times 10^{-2})$.

To compare numbers in scientific notation, first compare their powers of 10. The number with the larger exponent is greater. If the exponents are equal, compare the decimal coefficients.

Linear growth is additive, meaning a constant amount is added in each step. Exponential growth is multiplicative, meaning the quantity is multiplied by a constant factor in each step, leading to much faster increases.