What are exponents?
Exponents, besides known as powers, are values that show how many times to multiply a base act by itself. For example, 43 is telling you to multiply four by itself three times. 43= 4 × 4 × 4 = 64 The phone number being raised by a ability is known as the base, while the superscript number above it is the exponent or power .
Credit: To The Square Inch The equality above is said as “ four to the exponent of three ”. The power of two can besides be said as “ squared ” and the world power of three can be said as “ cubed ”. These terms are much used when finding the sphere or book of diverse shapes. Writing a number in exponential phase refers to simplifying it to a base with a might. For model, turning 5 × 5 × 5 into exponential form looks like 53. Exponents are a way to simplify equations to make them easier to read. This becomes particularly important when you ’ ra dealing with variables such as ‘ 𝒙 ’ and ‘ 𝑦 ’ — as 𝒙7× 𝑦5= ? is easier to read than (𝒙)(𝒙)(𝒙)(𝒙)(𝒙)(𝒙)(𝒙)(𝑦)(𝑦)(𝑦)(𝑦)(𝑦) = ?
Rules of exponents in everyday life
not entirely will understanding exponent properties help you to solve respective algebraic problems, exponents are besides used in a virtual manner in everyday life sentence when calculating square feet, square meters, and even cubic centimeters. advocate rules besides simplify calculating highly boastfully or highly bantam quantities. These are besides used in the global of computers and engineering when describing megabytes, gigabytes, and terabytes .
What are the different rules of exponents?
There are seven advocate rules, or laws of exponents, that your students need to learn. Each rule shows how to solve unlike types of mathematics equations and how to add, subtract, breed and separate exponents .
Make sure you go over each exponent dominion thoroughly in class, as each one plays an important function in solving advocate based equations .
1. Product of powers rule
When multiplying two bases of the lapp value, keep the bases the same and then add the exponents together to get the solution. 42× 45 = ? Since the base values are both four, keep them the lapp and then add the exponents ( 2 + 5 ) together. 42 × 45= 47 then multiply four by itself seven times to get the answer. 47 = 4 × 4 × 4 × 4 × 4 × 4 × 4 = 16,384 Let ’ s expand the above equality to see how this convention works :
In an equation like this, adding the exponents together is a shortcut to get the answer. here ’ s a more complicated doubt to try : ( 4𝒙2 ) ( 2𝒙3 ) = ? Multiply the coefficients together ( four and two ), as they are not the lapp base. then keep the ‘ 𝒙 ’ the same and add the exponents. ( 4𝒙2 ) ( 2𝒙3 ) = 8𝒙5
2. Quotient of powers rule
generation and division are opposites of each other — much the same, the quotient rule acts as the face-to-face of the product rule. When dividing two bases of the same value, keep the base the same, and then subtract the exponent values. 55 ÷ 53 = ? Both bases in this equality are five, which means they stay the lapp. then, take the exponents and subtract the divisor from the dividend. 55÷ 53 = 52 finally, simplify the equation if needed : 52= 5 × 5 = 25 once again, expanding the equality shows us that this shortcut gives the right answer :
Take a expression at this more complicate model : 5𝒙4 / 10𝒙2 = ? The like variables in the denominator cancel out those in the numerator. You can show your students this by crossing out an equal count of 𝒙 ’ sulfur from the circus tent and bed of the fraction. 5𝒙4 / 10𝒙2 = 5𝒙/10 then simplify where possible, as you would with any fraction. Five can go into ten-spot, five times turning the fraction into ½ with the remaining 𝒙 variables. 5𝒙4/10𝒙2= 1𝒙2/2 = 𝒙2/2
3. Power of a power rule
This convention shows how to solve equations where a power is being raised by another power. ( 𝒙3 ) 3 = ? In equations like the one above, multiply the exponents together and keep the base the same. ( 𝒙3 ) 3 = 𝒙9 Take a look at the expand equation to see how this works :
4. Power of a product rule
When any foundation is being multiplied by an advocate, distribute the advocate to each part of the basis. ( 𝒙𝑦 ) 3 = ? In this equation, the power of three needs to be distributed to both the 𝒙 and the 𝑦 variables. ( 𝒙𝑦 ) 3 = 𝒙3𝑦3
This rule applies if there are exponents attached to the base equally well. ( 𝒙2𝑦2 ) 3 = 𝒙6𝑦6 Expanded, the equality would look like this :
Both of the variables are squared in this equation and are being raised to the power of three. That means three is multiplied to the exponents in both variables turning them into variables that are raised to the power of six .
5. Power of a quotient rule
A quotient merely means that you ’ re dividing two quantities. In this principle, you ’ ra raising a quotient by a power. Like the power of a product dominion, the advocate needs to be distributed to all values within the brackets it ’ south attached to. ( 𝒙/𝑦 ) 4 = ? here, raise both variables within the brackets by the ability of four .
Take a look at this more complicate equality : ( 4𝒙3/5𝑦4 ) 2 = ? Don ’ metric ton forget to distribute the exponent you ’ re multiply by to both the coefficient and the variable. then simplify where possible. ( 4𝒙3/5𝑦4 ) 2= 42𝒙6/52𝑦8 = 16𝒙6/25𝑦8
6. Zero power rule
Any establish raised to the office of zero is equal to one .
The easiest manner to explain this rule is by using the quotient of powers rule. 43/43 = ? Following the quotient of powers rule, subtract the exponents from each other, which cancels them out, only leaving the basal. Any number divided by itself is one. 43/43= 4/4 = 1 No matter how long the equality, anything raised to the power of nothing becomes one. ( 82𝒙4𝑦6 ) 0 = ? typically, the away exponent would have to be multiplied throughout each phone number and variable star in the brackets. however, since this equation is being raised to the power of zero, these steps can be skipped and the suffice plainly becomes one. ( 82𝒙4𝑦6 ) 0 = 1 The equality fully expanded would look like this : ( 82𝒙4𝑦6 ) 0 = 80𝒙0𝑦0 = ( 1 ) ( 1 ) ( 1 ) = 1
7. Negative exponent rule
When there is a number being raised by a negative exponent, flip it into a reciprocal to turn the advocate into a positive. Don’t use the veto advocate to turn the base into a veto .
Credit: Thinglink We ’ ve talked about reciprocals before in our article, “ How to divide fractions in 3 easy steps ”. basically, reciprocals are what you multiply a number by to get the value of one. For example, to turn two into one, multiply it by ½. now, look at this exponent model : 𝒙-2 = ? To make a total into a reciprocal :
- Turn the number into a fraction (put it over one)
- Flip the numerator into the denominator and vice versa
- When a negative number switches places in a fraction it becomes a positive number
The finish of equations with negative exponents is to make them positive. immediately, take a front at this more complicate equation : 4𝒙-3𝑦2/20𝒙𝑧-3 = ? In this equation, there are two exponents with negative powers. Simplify what you can, then flip the negative exponents into their reciprocal imprint. In the solution, 𝒙-3 moves to the denominator, while 𝑧-3 moves to the numerator. Since there is already an 𝒙 respect in the denominator, 𝒙3 adds to that value. 4𝒙-3𝑦2/20𝒙z-3 = 𝑦2𝑧3/5𝒙4 With these seven rules in your students ’ back pockets, they ’ ll be able to take on most advocate questions they come across !
Exponent rules chart
How Prodigy can help you teach exponent rules
Prodigy is a curriculum-aligned mathematics game you can use to assign questions, track advance, and identify perturb spots in your students ’ learning. And you can create teacher and student accounts for free ! With sol many different advocate rules to follow and several students to track, it can be hard to see who needs assistant with what. Prodigy makes it easily to track progress, and create a alone bet on experience for each scholar based on their needs .
Statistics are tracked live, as students play the game, and feedback is available instantaneously. Most of the fourth dimension your students won ’ t even realize that they ’ re taking separate in mathematics lessons. It ’ s all separate of their individualized gambling experience ! From the teacher dashboard, you can create example plans, see bouncy statistics, input custom assignments, and prepare your students for approaching tests. Here’s how you can use Prodigy to :
- Prepare students for standardized tests
- Reinforce in-class concepts (like exponent rules)
- Differentiate math practice in the math classroom and at home
Free exponent rules worksheet
Math worksheets are handy tools that can show how students are understanding key concepts. You can see how students are coming up with answers, where they ’ rhenium struggling, and if any concepts need to be covered in more contingent. We ’ ve put together an advocate rules worksheet, with the aid of our team of teachers, to help you with exponent lessons. Click here to download our exponent rules worksheet, complete with an answer key !
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Conclusion: exponent rules practice
Exponents are used to show how many times a infrastructure prize is multiplied by itself. This simplifies equations to an easier to read format. ( 𝒙𝒙𝒙𝒙𝒙𝒙𝒙𝒙𝒙 ) ( 𝑦𝑦𝑦𝑦𝑦𝑦 ) ( 𝑧𝑧𝑧𝑧𝑧 ) = 𝒙9𝑦6𝑧5 To recap, there are seven basic rules that explain how to solve most mathematics equations that involve exponents. The exponent rules are :
- Product of powers rule — Add powers together when multiplying like bases
- Quotient of powers rule — Subtract powers when dividing like bases
- Power of powers rule — Multiply powers together when raising a power by another exponent
- Power of a product rule — Distribute power to each base when raising several variables by a power
- Power of quotient rule — Distribute power to all values in a quotient
- Zero power rule — Any base raised to the power of zero becomes one
- Negative exponent rule — To change a negative exponent to a positive one, flip it into a reciprocal
Exponents have a inclination of appearing throughout our lives, so it ’ mho crucial that students understand how they work moving forward. There are a draw of rules to remember but, once your students understand them, solving exponents will likely get easier !
