Numbers and operations: FAQ

Repeating decimals are when a pattern of numbers repeats over and over after the decimal point. For example, the fraction $\dfrac13$start fraction, 1, divided by, 3, end fraction is equivalent to the decimal $0.\overline{3}$0, point, start overline, 3, end overline, which is a repeating decimal.

The bar over the $3$3 means that it repeats forever: $0.\overline{3}=0.333333\dots$0, point, start overline, 3, end overline, equals, 0, point, 333333, dots

Square roots and cube roots are used all the time in math and science. When we want to find the side length of a square with a given area, we use square roots. When we want to find the length of a cube with a given volume, we use cube roots.

Irrational numbers are numbers that can't be written as a fraction of two integers. For example, $\sqrt{2}$square root of, 2, end square root is an irrational number, because no matter how hard we try, we can't find two integers that will give us $\sqrt{2}$square root of, 2, end square root when we divide them.

When we approximate an irrational number, we're finding a rational number that is close to the irrational number. For example, we could say that $\sqrt{2}\approx1.414$square root of, 2, end square root, approximately equals, 1, point, 414.

Exponent properties are rules that we can use to simplify expressions that contain exponents.

Product rule: $x^a \times x^b = x^{a+b}$x, start superscript, a, end superscript, times, x, start superscript, b, end superscript, equals, x, start superscript, a, plus, b, end superscript. For example, $x^2 \times x^3 = x^5$x, squared, times, x, cubed, equals, x, start superscript, 5, end superscript.

Power rule: $(x^a)^b = x^{ab}$left parenthesis, x, start superscript, a, end superscript, right parenthesis, start superscript, b, end superscript, equals, x, start superscript, a, b, end superscript. For example, $(x^2)^3 = x^6$left parenthesis, x, squared, right parenthesis, cubed, equals, x, start superscript, 6, end superscript.

Quotient rule: $\dfrac{x^a}{x^b} = x^{a-b}$start fraction, x, start superscript, a, end superscript, divided by, x, start superscript, b, end superscript, end fraction, equals, x, start superscript, a, minus, b, end superscript. For example, $\dfrac{x^5}{x^2} = x^3$start fraction, x, start superscript, 5, end superscript, divided by, x, squared, end fraction, equals, x, cubed.

Zero exponent rule: $x^0 = 1$x, start superscript, 0, end superscript, equals, 1. For example, $7^0 = 1$7, start superscript, 0, end superscript, equals, 1.

Scientific notation is a way of writing really big or really small numbers in a way that makes them easier to work with. For example, $0.000000000005$0, point, 000000000005 is hard to read, but when we put it in scientific notation, we get $5\times10^{-12}$5, times, 10, start superscript, minus, 12, end superscript. This is much easier to read and work with.

Scientific notation is used in many different fields, especially in the sciences. Scientists often work with very large or very small numbers, and using scientific notation makes calculations and comparisons much easier.

For example, in chemistry, scientists use scientific notation to express the mass of atoms and molecules in grams, which usually have values much smaller than $1$1.

Another example: we might want to calculate the distance that light travels in one year. We can use scientific notation to express the speed of light as $3 \times 10^8$3, times, 10, start superscript, 8, end superscript meters per second and multiply that by the number of seconds in a year to get our answer.