Exponents and scientific notation

An exponent is a number that indicates how many times to multiply the base by itself. For example, in $$2^3$$ , 2 is the base and 3 is the exponent, meaning $$2 \times 2 \times 2 = 8$$ .

Scientific notation is a way of expressing numbers that are too large or too small in a compact form. It is written as $$a \times 10^n$$ , where 1 ≤ a < 10 and n is an integer.

To add numbers in scientific notation, they must have the same exponent. If they do not, convert them to the same exponent before adding the coefficients.

When multiplying numbers in scientific notation, multiply the coefficients and add the exponents. For example, $$(2 \times 10^3) \times (3 \times 10^4) = 6 \times 10^{3+4} = 6 \times 10^7$$ .

When dividing numbers in scientific notation, divide the coefficients and subtract the exponents. For example, $$\frac{6 \times 10^7}{2 \times 10^3} = 3 \times 10^{7-3} = 3 \times 10^4$$ .

To convert a number from standard form to scientific notation, move the decimal point to create a number between 1 and 10, and count the number of places moved to determine the exponent.

$$\frac{a^7}{a^{14}} = a^{7-14} = a^{-7} = \frac{1}{a^7}$$ .

To find the difference, subtract Mercury's distance from Neptune's distance. If Mercury is $$5.7 \times 10^7$$ km and Neptune is $$4.5 \times 10^9$$ km, the difference is $$4.5 \times 10^9 - 5.7 \times 10^7$$ .

Convert $$13 \times 10^7$$ to $$1.3 \times 10^8$$ , then add: $$12 \times 10^6 + 1.3 \times 10^8 = 1.42 \times 10^8$$ .

Total weight = $$30 \times 3.8 \times 10^5 = 114 \times 10^5 = 1.14 \times 10^7$$ pounds.