What are exponents?
Exponents room numbers that have been multiply by themselves. For instance, 3 · 3 · 3 · 3 can be composed as the exponent 34: the number 3 has actually been multiplied by chin 4 times.
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Exponents room useful due to the fact that they let united state write lengthy numbers in a to rstarrkingschool.netce form. For instance, this number is really large:
But you can write it this method as an exponent:
It also works for little numbers with numerous decimal places. For instance, this number is very tiny but has numerous digits:
It also could be composed as one exponent:
Scientists often use exponents to convey very large numbers and also very little ones. You'll check out them regularly in algebra problems too.Understanding exponents
As you observed in the video, exponents space written favor this: 43 (you'd read it together 4 come the third power). All exponents have two parts: the base, i m sorry is the number gift multiplied; and the power, i beg your pardon is the number of times you multiply the base.
Because our base is 4 and our strength is 3, we’ll have to multiply 4 by chin three times.
43 = 4 ⋅ 4 ⋅ 4 = 64
Because 4 · 4 · 4 is 64, 43 is same to 64, too.
Occasionally, you might see the exact same exponent written prefer this: 5^3. Don’t worry, it’s precisely the same number—the base is the number come the left, and also the power is the number come the right. Depending upon the type of calculator girlfriend use—and especially if you’re utilizing the calculator on your phone or computer—you may need come input the exponent this method to calculate it.Exponents to the 1st and 0th power
How would you simplify these exponents?
Don’t feel poor if you’re confused. Even if you feel comfortable with other exponents, it’s not evident how to calculation ones through powers that 1 and also 0. Luckily, these exponents follow an easy rules:Exponents v a strength of 1Any exponent v a strength of 1 equals the base, for this reason 51 is 5, 71 is 7, and also x1 is x.Exponents with a strength of 0Any exponent through a strength of 0 equals 1, therefore 50 is 1, and also so is 70, x0, and any various other exponent v a power of 0 you have the right to think of.
Operations v exponents
How would you resolve this problem?
22 ⋅ 23
If friend think you must solve the index number first, climate multiply the resulting numbers, you’re right. (If you weren’t sure, inspect out our lesson on the order of operations).
How around this one?
x3 / x2
Or this one?
2x2 + 2x2
While girlfriend can’t exactly solve these problems without much more information, you have the right to simplify them. In algebra, girlfriend will regularly be asked to do calculations top top exponents through variables together the base. Fortunately, it’s straightforward to add, subtract, multiply, and also divide this exponents.Adding exponents
When you’re including two exponents, girlfriend don’t include the actual powers—you include the bases. Because that instance, to leveling this expression, you would certainly just include the variables. You have actually two xs, which deserve to be composed as 2x. So, x2+x2 would be 2x2.
x2 + x2 = 2x2
How around this expression?
3y4 + 2y4
You're including 3y come 2y. Since 3 + 2 is 5, that method that 3y4 + 2y4 = 5y4.
3y4 + 2y4 = 5y4
You can have noticed the we only looked at difficulties where the exponents us were including had the exact same variable and also power. This is due to the fact that you have the right to only add exponents if your bases and also exponents space exactly the same. So girlfriend can include these below since both terms have actually the same variable (r) and also the same power (7):
4r7 + 9r7
You can never add any that these as they’re written. This expression has variables v two different powers:
4r3 + 9r8
This one has actually the exact same powers but different variables, so girlfriend can't include it either:
4r2 + 9s2Subtracting exponents
Subtracting exponents works the exact same as including them. Because that example, can you figure out exactly how to simplify this expression?
5x2 - 4x2
5-4 is 1, therefore if you said 1x2, or simply x2, you’re right. Remember, just like with including exponents, you have the right to only subtract exponents through the same power and base.
5x2 - 4x2 = x2Multiplying exponents
Multiplying exponents is simple, but the way you carry out it could surprise you. To multiply exponents, add the powers. For instance, take this expression:
x3 ⋅ x4
The powers space 3 and 4. Due to the fact that 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, for this reason our streamlined exponent will have a power of 8. In this case, we’ll additionally need to main point the coefficients. The coefficients space 3 and also 2. We should multiply these choose we would any type of other numbers. 3⋅2 is 6, for this reason our streamlined answer is 6x8.
3x2 ⋅ 2x6 = 6x8
You deserve to only leveling multiplied exponents v the same variable. Because that example, the expression 3x2⋅2x3⋅4y2 would certainly be simplified to 24x5⋅y2. For much more information, go to our Simplifying expressions lesson.Dividing exponents
Dividing index number is similar to multiplying them. Rather of adding the powers, friend subtract them. Take this expression:
x8 / x2
Because 8 - 2 is 6, we understand that x8/x2 is x6.
x8 / x2 = x6
What about this one?
10x4 / 2x2
If girlfriend think the answer is 5x2, you’re right! 10 / 2 gives us a coefficient of 5, and also subtracting the strength (4 - 2) means the power is 2.Raising a strength to a power
Sometimes you could see an equation choose this:
An exponent on an additional exponent could seem confusing at first, but you already have every the skills you must simplify this expression. Remember, one exponent method that you're multiplying the base by chin that numerous times. For example, 23 is 2⋅2⋅2. The means, we have the right to rewrite (x5)3 as:
To multiply exponents through the same base, simply add the exponents. Therefore, x5⋅x5⋅x5 = x5+5+5 = x15.
There's actually an also shorter method to simplify expressions like this. Take one more look in ~ this equation:
(x5)3 = x15
Did you an alert that 5⋅3 also equals 15? Remember, multiplication is the very same as including something an ext than once. That way we can think the 5+5+5, which is what we did earlier, together 5 times 3. Therefore, once you advanced a power to a power you can multiply the exponents.
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Let's look at one an ext example:
Since 6⋅4 = 24, (x6)4 = x24
Let's look at one an ext example:
First, we deserve to rewrite this as:
Remember in multiplication, order does no matter. Therefore, we have the right to rewrite this again as:
Since 3⋅3⋅3⋅3 = 81 and also x8⋅x8⋅x8⋅x8 = x32, ours answer is:
Notice this would certainly have additionally been the exact same as 34⋅x32.
Still confused about multiplying, dividing, or elevating exponents to a power? check out the video clip below to discover a trick because that remembering the rules: