What I have learned about grade 9 exponents

what is an exponent?

an exponent is that little number above a normal number or letter.

$5^2$

(In this example 2 is the exponent)

What do exponents do?

exponents are like copiers they make math more simple and compact

they make it easier to write 7x7x7x7x7x7 by turning it into $7^6$

Forms of exponents

Expanded form: $3\cdot 3\cdot 3\cdot3$

Simplified form: $3^4$

evaluated form: $81$

(the example is three to the power of four but that could be any number to the power of any number.)

exponents of zero

exponents can be zero and there are many reasons you could have got one but if you do have a number with the exponent of zero than its going to equal one every time and it doesn’t matter if the base is two million it’s always going to be one.

negative bases

There are two types of negative bases when working with exponents there this type

$-3^2$

and this type

$(-3)^2$

There is no wrong type they just mean different things for example the first type means.

$- (3\cdot 3)$ = $-9$

and the second type means.

$-3\cdot -3$ = $9$

To help you with this

if the negative sign is in the brackets look for the exponent. If it’s a even number then the answer will be positive. If it’s not even then it will be negative. If there is a negative outside the brackets as well then the sign of whether it’s positive or not will change to the opposite.

If the negative sign is outside in the brackets then that means the final answer for that will be negative unless there is a negative inside the brackets as well and the exponent is an odd number.

What is the difference between evaluating and simplifying?

Simplifying is when you make the answer into a power instead of getting a precise number.

ex: $5^4\cdot 5^3$ = $5^7$

Evaluating is when you get a precise answer instead of being lazy.

ex: $2^4$ = 2x2x2x2 = 4×4 = 16

The multiplication law

Multiplying with exponents isn’t what you would expect. When multiplying with exponents you are actually adding because exponents are lazy!

ex: $2^3\cdot 2^5$ = $2^8$

(But this only works when the thing being copied is the same for both exponents.)

The division law

The division law is similar to the multiplication law but instead of adding the exponents we subtract the exponents.

ex: $5^9\div 5^4$ = $5^5$

(But this only works when the thing being copied is the same for both exponents.)

Power of a power law

This law is a bit complicated to explain but pretty simple. Basically exponents can have exponents like a layered cake.

ex: $(6^4)^3$

For this question you need to think about exponents and how they copy everything they are told to so that meant this question is.

(6x6x6x6)(6x6x6x6)(6x6x6x6) which is $6^12$

If you haven’t noticed this yet the exponent just multiplies all of the exponents in the brackets by itself.

$(6+4^2)^2$ = $6^2+4^4$

Now you might be saying yeah but in this equation six doesn’t have an exponent, why is it being effected? Before you ask that’s because six has an invisible exponent witch is one. In fact every number that doesn’t already have an exponent has an invisible exponent of one, so this equation is really.

$(6^1+4^2)^2$ = $6^2+4^4$

Exponents on variables

exponents on variable are very similar but have a few conditions. For instance you can still use the three laws above on the variables with exponents but that’s it. The variable numbers must stay in either expanded form or simplified form if you are not able to get the variable to an exponent of 0.

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