Squaring a base (raising a number to the power of 2) and taking the square root are similar concepts, many people consider one the opposite or the undoing of the other. If you want to square the number 6, you take 6 * 6 = 36. Now if you want to find what two identical numbers multiply to give you 36, you take the square root of 36. This square root gives the value of 6. It can also be noted that squaring a square root removes the radical. Show Likewise, cubing a base (raising a number to the power of 3) will give us a perfect cube. In case you need to calculate the cube root you can use our cube root calculator which is an excellent tool that will calculate the cube root of any number. In modular arithmetic there are dedicated methods of exponentiation - learn more with the power mod calculator. Besides, you may check our logarithm calculator which is the inverse function of the exponent. Any number raised to the power of 0 equals 1. The negative exponent calculator is useful when dealing with exponential decay, which has a negative exponent in its formula.
 The exponent of a number says how many times to use the number in a multiplication. In 82 the "2" says to use 8 twice in a multiplication, In words: 82 could be called "8 to the power 2" or "8 to the second power", or simply "8 squared" Exponents are also called Powers or Indices. Some more examples: Example: 53 = 5 × 5 × 5 = 125
Example: 24 = 2 × 2 × 2 × 2 = 16
Exponents make it easier to write and use many multiplications
Example: 96 is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9 You can multiply any number by itself as many times as you want using exponents. Try here: algebra/images/exponent-calc.js So in general:
Another Way of Writing ItSometimes people use the ^ symbol (above the 6 on your keyboard), as it is easy to type.
Example: 2^4 is the same as 24 Negative ExponentsNegative? What could be the opposite of multiplying? Dividing! So we divide by the number each time, which is the same as multiplying by 1number
Example: 8-1 = 18 = 0.125 We can continue on like this:
Example: 5-3 = 15 × 15 × 15 = 0.008 But it is often easier to do it this way:
5-3 could also be calculated like: 15 × 5 × 5 = 153 = 1125 = 0.008 Negative? Flip the Positive!
More Examples:
What if the Exponent is 1, or 0?
It All Makes SenseIf you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern:
Be Careful About GroupingTo avoid confusion, use parentheses () in cases like this:
305, 1679, 306, 1680, 1077, 1681, 1078, 1079, 3863, 3864 Copyright © 2021 MathsIsFun.com
/en/algebra-topics/order-of-operations/content/ What are exponents?Exponents are numbers that have been multiplied by themselves. For instance, 3 · 3 · 3 · 3 could be written as the exponent 34: the number 3 has been multiplied by itself 4 times. Exponents are useful because they let us write long numbers in a shortened form. For instance, this number is very large: 1,000,000,000,000,000,000 But you could write it this way as an exponent: 1018 It also works for small numbers with many decimal places. For instance, this number is very small but has many digits: .00000000000000001 It also could be written as an exponent: 10-17 Scientists often use exponents to convey very large numbers and very small ones. You'll see them often in algebra problems too. Understanding exponentsAs you saw in the video, exponents are written like this: 43 (you'd read it as 4 to the 3rd power). All exponents have two parts: the base, which is the number being multiplied; and the power, which is the number of times you multiply the base. Because our base is 4 and our power is 3, we’ll need to multiply 4 by itself three times. 43 = 4 ⋅ 4 ⋅ 4 = 64 Because 4 · 4 · 4 is 64, 43 is equal to 64, too. Occasionally, you might see the same exponent written like this: 5^3. Don’t worry, it’s exactly the same number—the base is the number to the left, and the power is the number to the right. Depending on the type of calculator you use—and especially if you’re using the calculator on your phone or computer—you may need to input the exponent this way to calculate it. Exponents to the 1st and 0th powerHow would you simplify these exponents? 71 70 Don’t feel bad if you’re confused. Even if you feel comfortable with other exponents, it’s not obvious how to calculate ones with powers of 1 and 0. Luckily, these exponents follow simple rules:
Operations with exponentsHow would you solve this problem? 22 ⋅ 23 If you think you should solve the exponents first, then multiply the resulting numbers, you’re right. (If you weren’t sure, check out our lesson on the order of operations). How about this one? x3 / x2 Or this one? 2x2 + 2x2 While you can’t exactly solve these problems without more information, you can simplify them. In algebra, you will often be asked to perform calculations on exponents with variables as the base. Fortunately, it’s easy to add, subtract, multiply, and divide these exponents. Adding exponentsWhen you’re adding two exponents, you don’t add the actual powers—you add the bases. For instance, to simplify this expression, you would just add the variables. You have two xs, which can be written as 2x. So, x2+x2 would be 2x2. x2 + x2 = 2x2 How about this expression? 3y4 + 2y4 You're adding 3y to 2y. Since 3 + 2 is 5, that means that 3y4 + 2y4 = 5y4. 3y4 + 2y4 = 5y4 You might have noticed that we only looked at problems where the exponents we were adding had the same variable and power. This is because you can only add exponents if their bases and exponents are exactly the same. So you can add these below because both terms have the same variable (r) and the same power (7): 4r7 + 9r7 You can never add any of these as they’re written. This expression has variables with two different powers: 4r3 + 9r8 This one has the same powers but different variables, so you can't add it either: 4r2 + 9s2 Subtracting exponentsSubtracting exponents works the same as adding them. For example, can you figure out how to simplify this expression? 5x2 - 4x2 5-4 is 1, so if you said 1x2, or simply x2, you’re right. Remember, just like with adding exponents, you can only subtract exponents with the same power and base. 5x2 - 4x2 = x2 Multiplying exponentsMultiplying exponents is simple, but the way you do it might surprise you. To multiply exponents, add the powers. For instance, take this expression: x3 ⋅ x4 The powers are 3 and 4. Because 3 + 4 is 7, we can simplify this expression to x7. x3 ⋅ x4 = x7 What about this expression? 3x2 ⋅ 2x6 The powers are 2 and 6, so our simplified exponent will have a power of 8. In this case, we’ll also need to multiply the coefficients. The coefficients are 3 and 2. We need to multiply these like we would any other numbers. 3⋅2 is 6, so our simplified answer is 6x8. 3x2 ⋅ 2x6 = 6x8 You can only simplify multiplied exponents with the same variable. For example, the expression 3x2⋅2x3⋅4y2 would be simplified to 24x5⋅y2. For more information, go to our Simplifying Expressions lesson. Dividing exponentsDividing exponents is similar to multiplying them. Instead of adding the powers, you subtract them. Take this expression: x8 / x2 Because 8 - 2 is 6, we know that x8/x2 is x6. x8 / x2 = x6 What about this one? 10x4 / 2x2 If you think the answer is 5x2, you’re right! 10 / 2 gives us a coefficient of 5, and subtracting the powers (4 - 2) means the power is 2. Raising a power to a powerSometimes you might see an equation like this: (x5)3 An exponent on another exponent might seem confusing at first, but you already have all the skills you need to simplify this expression. Remember, an exponent means that you're multiplying the base by itself that many times. For example, 23 is 2⋅2⋅2. That means, we can rewrite (x5)3 as: x5⋅x5⋅x5 To multiply exponents with the same base, simply add the exponents. Therefore, x5⋅x5⋅x5 = x5+5+5 = x15. There's actually an even shorter way to simplify expressions like this. Take another look at this equation: (x5)3 = x15 Did you notice that 5⋅3 also equals 15? Remember, multiplication is the same as adding something more than once. That means we can think of 5+5+5, which is what we did earlier, as 5 times 3. Therefore, when you raise a power to a power you can multiply the exponents. Let's look at one more example: (x6)4 Since 6⋅4 = 24, (x6)4 = x24 x24 Let's look at one more example: (3x8)4 First, we can rewrite this as: 3x8⋅3x8⋅3x8⋅3x8 Remember in multiplication, order does not matter. Therefore, we can rewrite this again as: 3⋅3⋅3⋅3⋅x8⋅x8⋅x8⋅x8 Since 3⋅3⋅3⋅3 = 81 and x8⋅x8⋅x8⋅x8 = x32, our answer is: 81x32 Notice this would have also been the same as 34⋅x32. Still confused about multiplying, dividing, or raising exponents to a power? Check out the video below to learn a trick for remembering the rules: /en/algebra-topics/negative-numbers/content/ |
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