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Exponents commonly confuse uniformed mathematicians with their complex algorithms and formulas. But, you, the reader, don't have to be defeated by this seemingly impossible to comprehend mathmatical term. Lucky for you, this website will guide you towards the path of everlasting knowledge of exponents. When the end of this website is reached, you will completely understand what exponents are, the multiplication and division of exponents, and negative exponents. There are many intricate rules and loopholes exponents conceal, and it's about time you understand and learn them. Through this website, you'll finally unravel the secrets exponents hold... Ready to embark on this adventure?
Let's tackle the multiplication of powers/exponents.
Luckily, there is a rule called the exponent "product rule" that can help us multiply exponents,
powers. The rule states that when two powers with equal bases are being multiplied together,
you can simply add the exponents and retain the same base. The formula for this is shown above.
Exponents are a condesed way to represent the amount of times a number, refered to as the base, is being multiplied by itself.
The amount of times the base is being multiplied by itself is the exponent.
If you are referring to the "whole" expression (the result of the base raised to the power), you would call it a power. Powers are expressed as the base number "raised to the power of" the exponent number.
For example in the picture above,
it shows 3 being multiplied by itself twice, so 3*3.
3 is the base in this example because 3 is the number being multiplied.
Since there are two 3's being multiplied together, the exponent is 2, as shown above.
3 to the second power, or 1*3*3, equals 9.
Written out powers (like 3*3) are always multiplied by 1.
In the digital world, powers are written as b^p (base raised to the power of the exponent).
The ^ (caret) shows the following number is considered the exponent.
In mathmatics, exponents follow a reliable set of rules and patterns. Here are some of the more important and common ones...
First, any number raised to the zeroth power (b^0) equals 1 since 3 multiplied by itself zero times equals nothing. So, 1 isn't being multiplied by any other number
Second, if an exponent has a negative base, the negative number has to be within closed parenthese for the exponent to apply to the negative term. So, -3^2 equals -(9), -(3*3), while (-3)^2 equals (9), (-3*-3).
Third, a negative base raised to an even power will result in a positive number, and a negative base raised to an odd power will result in a negative number.
Fourth, when raising a power to an exponent, all that is required is to multiply the two exponents together and retain the original base. E.g., (3^2)^2 equals 3^(2(2)) which equals 3^4, or 81. If you wrote out the entire expression instead, you would end up with the same output number, 81, since 3^2 equals 9. 9 raised to the 2nd power equals 81. 81 equals 81.
Fifth rule, exponents can be canceled out, just like addition and subtraction. Radicals cancel out exponents. In case you aren't aware, a radical is any expresion that includes a radical symbol (√). This symbol, when placed before a value or number, represents the root of that quantity, most commonly the square root. To indicate roots higher than square roots, a raised number would be shown above the radical. Sometimes, a variable will be raised to power, and the only way to isolate said variable is to root the variable with the corresponding root. For example, if the variable x was raised to the 3rd power, a cube root (a radical with a raised 3) would cancel out the exponent.
Lastly, 1 raised to any power always equals 1 because no matter how many times you multiply one by itself, it'll always equal 1.
As you may know, negative means the opposite of what directly follows the negative symbol. That pattern still stays true even with powers. Positive exponents represents the amount of times the base is being multiplied. But negative exponents represent the amount of times the number 1 is being divided by the base since powers are always secretly multiplied by 1 even if it's not exactly shown. Negative exponents are most commonly written like the image above, a fraction where 1 is the numerator and the power with the negative exponent is the denominator, but the exponent is now positive. Writing negative exponents in that form displays the exact same imformation i.e. 1 divided by the base raised to power.
There are no extra "rules" that only apply to negative exponents. Every rule or property that applies to powers with a positive exponent also applies to powers with a negative exponent, except for the rule shown in the previous section. For example, if you were to multiply powers with negative exponents, you would still add the two bases together, although in hindsight you would be subtracting the two.
Also, we should learn about dividing powers/exponents.
Dividing powers is fairly similar to multiplying them. But, instead of adding the exponents, you
would subtract the divisor's exponent from the dividend's exponent. The formula for this is shown
above.