Can polynomials have negative exponents

ExponentsDealing with positive and negative exponents and simplifying expressions dealing with them is simply a matter of remembering what the definition of an exponent is. ∞A positive exponent means repeated multiplication. ∞A negative exponent means the opposite of repeated multiplication, which is repeated division. Let’s evaluate the following: 23 = 22 = 21 = 20 = 2 –1 = 2 –2 = 2 –3 = Notice that a negative exponent makes the “2” flip from12 raised to some power over to 21 raised to some power. In other words, instead of multiplying by 2 repeatedly, we’re dividing by two repeatedly. In effect, a negative exponent makes us do the opposite of what it initially looks like. If a number with a negative exponent is in the top (multiplication), it means to move it to the bottom (divide); If a number with a negative exponent is in the bottom (division), it means to move it to the top (multiply). Examples: Simplify the following. 1) x – 52) 6x1−3) 5x – 44) 2x65−5) (3x) –2Notice that only whatever is directly next to the exponent is affected by the exponent. In #3 and #4, the numbers 5 and 6 are not moved or otherwise changed. In #5, the entire parenthesis is affected by the exponent. Laws of ExponentsAt this point, the laws of exponents should be fairly well known to us. We’re going to examine where these “laws” came from. They’re just based on the definition of an exponent, which is repeated multiplication. Example 1: Simplify: 42xx•Example 2: Simplify 62xx
Example 3: Simplify ()43xExample 4: Simplify 234yxWhen simplifying an expression involving positive and negative exponents, it’s sometimes easier to move the base with the negative exponents before using the laws from Examples 1 and 2. That way, we don’t have to worry about double negatives and the like. I tend to distribute first (like in Examples 3 and 4), move my negative exponents, and then do the final simplification (Examples 1 and 2). Practice – Simplify.1) 54xx− 2) ()423yx−3) 623yx−−4) 28532yx4yx2−−−5) 24852x16y15y25x12•−−−Scientific NotationScientific notation is used by mathematicians, scientists, and many others whenever the numbers they are dealing with are either really, really large or really, really small. It prevents us from having to write a bunch of zeroes. Examples of numbers in scientific notation (and the equivalent expanded number): 6.0221415 × 10 23 (Avogadro's Constant) 602,214,150,000,000,000,000,000 1.98892 × 10 30 (mass of the sun in kg) 1,988,920,000,000,000,000,000,000,000,000 2.8 x 10 – 5 (mass of a grain of sand in g) 0.000028 2.5 x 10 – 2 (mass of a drop of water in g) 0.025 Notice that the first two numbers have positive exponents on the “10”, while the last two numbers have negative exponents on the “10”. This is because the first two are very large numbers (we’re multiplying by 10 repeatedly), whereas the last two are very small numbers (we’re dividing by 10 repeatedly).