A positive exponent indicates how many times a base number should be multiplied, whereas a negative exponent indicates how many times a base number should be divided. Negative exponents, such as x-n, can be rewritten as 1/1 xn. For instance, 2-4 = 1/16 (24) = 1/16 (2x) where x is the number used to raise 2 to the power of n. There are no rules on how to calculate **negative exponents**, so they must be calculated case by case.

- How do you do negative exponents?
- How do you calculate negative powers?
- Do negative exponents make numbers smaller?
- Can the power of e be negative?
- What does "negative exponent" mean?
- What is the inverse of a negative power?
- How do you get rid of three exponents?
- Is a negative number with an exponent positive?

A negative exponent indicates how many times the number should be divided by. That last example demonstrated a simpler technique to deal with **negative exponents**: Determine the positive exponent (an). The reciprocal (i.e., 1/an) is then calculated. This works because for **any real number** x, either x or 1/x is always positive.

There are two methods for calculating negative powers that differ only in their notation: The first method uses the fact that for any integer or fractional power, it can be rewritten as a positive power followed by negative power. For example, to find (-5) to the second power, multiply 5 by itself minus one time. So, 25-1=24.

The second method uses the fact that for any negative power, it can be rewritten as a positive power followed by a negative power. So, to find (-5) to **the second power** using **this method**, multiply 5 by itself plus one time. So, 25+1=26.

Either method can be used to solve problems involving negative powers. It just depends on what approach makes sense given the situation at hand.

Negative exponents are therefore defined as the positive reciprocal of the base multiplied by itself x times. The lower the integer, the bigger the negative exponent. For example, -5 is equivalent to 0.01 in **scientific notation**, while -50 is equal to 1 over 100000.

Because 7 to **the -3rd power** is approximately 5 x 10^-3, this means that 7 is a small number.

The reason that negative exponents make numbers appear smaller is simple arithmetic: If you take **any number** and multiply it by itself **some amount** less than one, then the result will be smaller than the original number.

For example, if I say that the number 7 has been multiplied by itself 3 times, then the answer is 21. Now let's say that instead we had done 72 times 7, which is about 513. This would have been a lot larger number than 721.

In general, any number raised to the power of another number less than one will give a result that is smaller than the original number.

The reciprocal is denoted by a negative sign on an exponent. Consider this: a positive exponent represents **repeated multiplication** by the base, whereas a negative exponent represents repeated division by the base. 2(-4) = 1/(24) = 1/(2*2*2*2) = 1/16 as a result. The correct answer is 1/16.

According to the Wikipedia, a negative exponent is just "the multiplicative inverse of the base increased to the positive opposite of the power." That's not very helpful! Let's try another source: The New York Times explains that a negative exponent means that "a left-hand side minus b right-hand side equals c."

That makes **more sense** to me. And here's how that works out mathematically: If you have a number with **a negative exponent**, then it turns out that the result is the same as if you had multiplied the number by **its additive inverse** instead. For example, if I say that 5^-1 = 23, then 5 * 22 = 120, which is the same thing as saying that 5^1 = 20 and 5^0 = 1. Multiplying by 0 doesn't change the value of a number, so we can cancel out the 0 from both sides and get 5 * 1 = 5, which is the same thing as saying that 5^-1 = 23.

That's exactly right!

A negative exponent is the multiplicative inverse of the base raised to the power opposite to the provided power. Expressions involving numbers with negative exponents

Negative Exponent | Result |
---|---|

3-2 | 1/32=1/9 |

x-3 | 1/x3 |

(2 + 4x)-2 | 1/(2+4x)2 |

(x2+ y2)-3 | 1/(x2+y2)3 |

An exponent denotes the number of times a number to be multiplied by itself. For example, x 3 (or x cubed) might be represented as x x x x x x x x x x x x x x x x x x x x x x x x To cancel out a component in an equation, use the inverse of that component. Subtraction, for example, eliminates the positive 4. Division by 2 reduces all the numbers by one half. Thus, division can be used to eliminate a component.

Three exponents are used in mathematics to denote a cube (or third power). That is, tripling (or multiplying by itself three times) a number. There are only two ways to triple a number: either repeat it twice or move **one copy** of it to the left and then another copy to the right. So there are only two ways to express 32 = 2 + 2 2 - 2 2 with exponents: 2^3 or 8. The same is true for 63 = 3 3 3 = 27 or 9. 98 = 2 2 - 1 -1 so 100000 = 88 = 7 2 7 2. Multiplying consecutive integers is easy- they just keep on adding up until there are no more, so the first step is to see how many times you can add 2 plus itself. In **this case** there are two copies of 2, so you can add them together four times; thus, the final answer is 64.

Exponents of **negative numbers** If the base is negative and the exponent is an even integer, the result is always positive. The final product will always be a negative number if the base is negative and the exponent is an odd integer.

If the base is negative and the exponent is not an integer, the result will be a negative number unless the exponent is zero or minus one. At **that point**, the result is positive infinity. For example, -5.55e-17 has the same value as 5.55e-17, and both are negative numbers.