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Understanding: Powers with the Same Exponent Video Lecture | Mathematics (Maths) Class 7
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FAQs on Understanding: Powers with the Same Exponent Video Lecture - Mathematics (Maths) Class 7
| 1. What is the definition of powers with the same exponent? |  |
Ans. Powers with the same exponent refer to numbers that are raised to the same power or have the same exponent. For example, if we have the numbers 2, 3, and 4, and we raise them to the power of 2, we get 2^2, 3^2, and 4^2 respectively. In this case, the exponent is 2, and the numbers have the same exponent.
| 2. How do you simplify powers with the same exponent? | |
Ans. To simplify powers with the same exponent, you need to apply the exponent to each number individually. For instance, if we have 2^3, 4^3, and 6^3, all raised to the power of 3, we simplify them as (2^3)^3, (4^3)^3, and (6^3)^3. Then, we can further simplify it as 2^(3*3), 4^(3*3), and 6^(3*3), which eventually results in 2^9, 4^9, and 6^9 respectively.
| 3. Can powers with the same exponent have different bases? | |
Ans. Yes, powers with the same exponent can have different bases. The exponent only affects the individual numbers or bases it is applied to. For example, if we have 2^4 and 3^4, both raised to the power of 4, they would simplify as 2^(4*4) and 3^(4*4), which further simplifies to 2^16 and 3^16 respectively.
| 4. What is the significance of powers with the same exponent in mathematics? | |
Ans. Powers with the same exponent are significant in mathematics as they allow us to compare and analyze numbers with a common power. It helps in understanding the relationships between different numbers and their exponential growth or decay. Powers with the same exponent also play a crucial role in various mathematical operations and formulas.
| 5. How can powers with the same exponent be used in real-life scenarios? | |
Ans. Powers with the same exponent can be used in real-life scenarios to analyze and compare quantities that undergo exponential growth or decay. For example, in finance, powers with the same exponent can be used to calculate compound interest over time. In science, they can be used to model population growth, radioactive decay, or the decay of certain substances. Additionally, powers with the same exponent are also used in fields such as engineering and computer science for various calculations and simulations.
Text Transcript from Video
hello friends this video on exponents
and powers part six is brought to you by
exam for calm no hole fear from exam
category now let's look at the next
category which is power to the power so
when you have a number a to the power M
which which in itself has a power and
you put another power to this number
that is a to the power M to the power n
so in that case the two these two powers
they get multiplied so this is equal to
a to the power M n let's take an example
let's say 2 to the power 3 this whole to
the power 2 so what is the meaning this
means that this entire number is
multiplied for 2 times right so what is
this entire number 2 into 2 into 2 and
this is multiplied for 2 times again so
- those are for 4 to the a so this means
a to the power 2 and a to the power 2
means 8 multiplied for 2 times so 8 into
8 is 64 so how can you write 64 in
exponential form again you need to find
out the prime factorization so let us do
the prime factorization here so 2 3 is
the 6 to do so for 2 16 to 32 2 8 16 to
fours are a two twos are four
so therefore 64 can be written as 2 into
2 into 2 into 2 so it gets repeated 4 6
times so this can be written as 2 to the
power 6 so this 6 can be written as 3
into 2 so when you look at it you see
that this power and this power that is 3
& 2
they get multiplied in the product so
therefore this holds true a to the power
n to the power n is equal to a to the
power M n now let us look at another
example
so let's take an exam
number two let's say you have something
like this - three - the power to this
whole to the power three so in this case
what would happen so - three square
means - 3 into minus 3 and this whole to
the power 3 so 3 3 is a 9 and 2 minus
sign this makes it positive so this is 9
to the power 3 that means 9 into 9 into
9 which is equal to 729 so in this case
also you do prime factorization of 729
so you find that it is exactly divisible
by 3 so 729 can be written as 3 into 3
into 3 3 3 3 that is 3 multiplied 4 6
times so this can be written as 3 to the
power 6 so this 6 can be written as 2
into 3 so that's how this also proves
that a to the power M to the power n is
equal to a to the power M in so let us
now move on to the third category where
we have powers with same exponent but
different bases so here also we will
look for both the multiplication and
division scenario so as per the law it
says that a to the power M multiplied by
B to the power M so you see a and B are
two different places but both of them
have the same exponent that is M so when
you multiply two different bases with
same exponents so in this case what will
happen now since the exponent is same so
obviously the product will also have the
same exponent so no new exponent will
come out of the way so the product would
also have the same exponent that is M
but what about the base now since here
we are trying to multiply these two
numbers therefore the base will be the
multiplication of the bases of the two
numbers so that's a simple way to
remember this now how do we prove that
yes this holds true so for that we take
example let's say 2 to the power 3
multiplied by 4 to the power 3 so you
have different bases 2 & 4 but you have
same exponent that is 3 so 2 to the
power 3 is 2
two into two and four to the power 3 is
4 into 4 into 4 so 2 into 2 into 2 is 8
and 400 4 into 4 is 64 4 4 16 16 into 4
is 64 so 8 into 64 is equal to 512 now
how do we write 512 in exponential form
because here right hand side we have
exponential form so for that we find out
factorization of 512 so we see that 512
is exactly divisible by 2 so 2 into 256
then 2 into 128 then 2 6.12 2 48 2 3 is
6 2 2 so 4 to 16 to 32 2 X is 16 2 4 z a
to 2 so 4 therefore 512 is equal to 1 2
3 4 5 6 7 8 9 2 multiply it by 9 dimes
right so this is equal to 2 to the power
9 so this 2 to the power 9 can also be
written like this now 2 into 2 into 2
what is 2 into 2 into 2 this is 8 so 8 x
8 again multiplied by 8 because to use
of for 48 so basically a 512 is also
equal to 8 to the power 3 so when you
look at this what do you see you see
that if you look at the product you see
that here the exponent is the same as
that of these two numbers so both the
numbers had 3 as exponent so the product
also has T as exponent but when you look
at the base you see that the base is
nothing but product of the two numbers
so this can be written as 2 into 4 to
the power 3 and that is what exactly
this law tells us so in a similar way
let us look at the division part as well
so we will take the same example that is
2 to the power 3 divided by
to the power 3 so this becomes equal to
2 into 2 into 2 divided by 4 into 4 into
4 so to use a 4 4 to say a and in the
denominator it would be 4 4 16 16 4 to
64 so 8 by 64 so now here you see 8 8 is
a 64 therefore this becomes equal to 1
by 8 so how can we write 1 so 1 can be
written as 1 to the power 3 and 8 can be
written as in to the power trillion in a
similar way by trying factorization so
from this we see that 1 by 2 to the
power 3 now 1 by 2 can be very well
written as 2 by 4 to the power 3 that's
because 2 by 4 is nothing but 1 by 2 so
forth right so so here when you look at
the product what do you see here also
the exponent is 3 which is the same as
the exponents of the two numbers and
when you look at the base so the bases
are just plain division of the business
of these 2 numbers that is 2 and 4 so
this proves that all the laws of
exponents are correct and they hold true
for all integers so let us try to solve
a few more examples using the laws of
exponents so let us look at the first
example 6 to the power 15 divided by 6
to the power 10 so here what do you see
you have the same base but different
exponents so what do we do when we have
same base and different exponent now the
base is the same so definitely the
product would also have the same base
that the 6 now what do we do with the
exponent what should be the exponent of
the result 15 or 10 so in this case it
is division between the two so what do
we do we just subtract the exponents so
this becomes 15 minus 10 is 5 so 6 to
the power 5 so that is 6 multiplied 5
times
and this value comes out to be seven
seven seven six so this becomes the
answer let us look at one more example
okay so I have written here so like let
me do it here 3 to the power 7 divided
by 3 to the power 4 into 3 to the power
3 so how do we solve this so in this
case again you see you have the same
things so everywhere you have a base 3
but you have different exponents perfect
now let us first handle this
multiplication part you remember the
board misrule where you first handled
the brackets then division then
multiplication in an addition and then
subtraction now in this case when you
see the denominator it is like a bracket
so you need to first sort out the
denominator part so what about the
denominator you have to same bases so
obviously the product would be also the
same base that is 3 what about the
powers the powers will get added because
here it is multiplication so the power
would get added so this is 3 to the
power 7 divided by 3 to the power 7 so
now again you have two numbers with the
same base so that means the product will
again have the same base 3 but what do
you do with the powers powers will get
subtracted because in this case it is
division so 7 minus 7 so 7 minus 7 is 0
so this is 3 to the power 0 anything to
the power 0 any integer to the power 0
is equal to 1 so therefore the result
would be 1 thank you please visit exam
fee or come for free quality education
you can learn with a simple four-step
learning process wherein you can watch
video lessons you can ask your questions
you can refer notes and you can take a
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class 6 to 12 on physics chemistry
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practical videos so please subscribe to
our channel for daily updates thank you
and powers part six is brought to you by
exam for calm no hole fear from exam
category now let's look at the next
category which is power to the power so
when you have a number a to the power M
which which in itself has a power and
you put another power to this number
that is a to the power M to the power n
so in that case the two these two powers
they get multiplied so this is equal to
a to the power M n let's take an example
let's say 2 to the power 3 this whole to
the power 2 so what is the meaning this
means that this entire number is
multiplied for 2 times right so what is
this entire number 2 into 2 into 2 and
this is multiplied for 2 times again so
- those are for 4 to the a so this means
a to the power 2 and a to the power 2
means 8 multiplied for 2 times so 8 into
8 is 64 so how can you write 64 in
exponential form again you need to find
out the prime factorization so let us do
the prime factorization here so 2 3 is
the 6 to do so for 2 16 to 32 2 8 16 to
fours are a two twos are four
so therefore 64 can be written as 2 into
2 into 2 into 2 so it gets repeated 4 6
times so this can be written as 2 to the
power 6 so this 6 can be written as 3
into 2 so when you look at it you see
that this power and this power that is 3
& 2
they get multiplied in the product so
therefore this holds true a to the power
n to the power n is equal to a to the
power M n now let us look at another
example
so let's take an exam
number two let's say you have something
like this - three - the power to this
whole to the power three so in this case
what would happen so - three square
means - 3 into minus 3 and this whole to
the power 3 so 3 3 is a 9 and 2 minus
sign this makes it positive so this is 9
to the power 3 that means 9 into 9 into
9 which is equal to 729 so in this case
also you do prime factorization of 729
so you find that it is exactly divisible
by 3 so 729 can be written as 3 into 3
into 3 3 3 3 that is 3 multiplied 4 6
times so this can be written as 3 to the
power 6 so this 6 can be written as 2
into 3 so that's how this also proves
that a to the power M to the power n is
equal to a to the power M in so let us
now move on to the third category where
we have powers with same exponent but
different bases so here also we will
look for both the multiplication and
division scenario so as per the law it
says that a to the power M multiplied by
B to the power M so you see a and B are
two different places but both of them
have the same exponent that is M so when
you multiply two different bases with
same exponents so in this case what will
happen now since the exponent is same so
obviously the product will also have the
same exponent so no new exponent will
come out of the way so the product would
also have the same exponent that is M
but what about the base now since here
we are trying to multiply these two
numbers therefore the base will be the
multiplication of the bases of the two
numbers so that's a simple way to
remember this now how do we prove that
yes this holds true so for that we take
example let's say 2 to the power 3
multiplied by 4 to the power 3 so you
have different bases 2 & 4 but you have
same exponent that is 3 so 2 to the
power 3 is 2
two into two and four to the power 3 is
4 into 4 into 4 so 2 into 2 into 2 is 8
and 400 4 into 4 is 64 4 4 16 16 into 4
is 64 so 8 into 64 is equal to 512 now
how do we write 512 in exponential form
because here right hand side we have
exponential form so for that we find out
factorization of 512 so we see that 512
is exactly divisible by 2 so 2 into 256
then 2 into 128 then 2 6.12 2 48 2 3 is
6 2 2 so 4 to 16 to 32 2 X is 16 2 4 z a
to 2 so 4 therefore 512 is equal to 1 2
3 4 5 6 7 8 9 2 multiply it by 9 dimes
right so this is equal to 2 to the power
9 so this 2 to the power 9 can also be
written like this now 2 into 2 into 2
what is 2 into 2 into 2 this is 8 so 8 x
8 again multiplied by 8 because to use
of for 48 so basically a 512 is also
equal to 8 to the power 3 so when you
look at this what do you see you see
that if you look at the product you see
that here the exponent is the same as
that of these two numbers so both the
numbers had 3 as exponent so the product
also has T as exponent but when you look
at the base you see that the base is
nothing but product of the two numbers
so this can be written as 2 into 4 to
the power 3 and that is what exactly
this law tells us so in a similar way
let us look at the division part as well
so we will take the same example that is
2 to the power 3 divided by
to the power 3 so this becomes equal to
2 into 2 into 2 divided by 4 into 4 into
4 so to use a 4 4 to say a and in the
denominator it would be 4 4 16 16 4 to
64 so 8 by 64 so now here you see 8 8 is
a 64 therefore this becomes equal to 1
by 8 so how can we write 1 so 1 can be
written as 1 to the power 3 and 8 can be
written as in to the power trillion in a
similar way by trying factorization so
from this we see that 1 by 2 to the
power 3 now 1 by 2 can be very well
written as 2 by 4 to the power 3 that's
because 2 by 4 is nothing but 1 by 2 so
forth right so so here when you look at
the product what do you see here also
the exponent is 3 which is the same as
the exponents of the two numbers and
when you look at the base so the bases
are just plain division of the business
of these 2 numbers that is 2 and 4 so
this proves that all the laws of
exponents are correct and they hold true
for all integers so let us try to solve
a few more examples using the laws of
exponents so let us look at the first
example 6 to the power 15 divided by 6
to the power 10 so here what do you see
you have the same base but different
exponents so what do we do when we have
same base and different exponent now the
base is the same so definitely the
product would also have the same base
that the 6 now what do we do with the
exponent what should be the exponent of
the result 15 or 10 so in this case it
is division between the two so what do
we do we just subtract the exponents so
this becomes 15 minus 10 is 5 so 6 to
the power 5 so that is 6 multiplied 5
times
and this value comes out to be seven
seven seven six so this becomes the
answer let us look at one more example
okay so I have written here so like let
me do it here 3 to the power 7 divided
by 3 to the power 4 into 3 to the power
3 so how do we solve this so in this
case again you see you have the same
things so everywhere you have a base 3
but you have different exponents perfect
now let us first handle this
multiplication part you remember the
board misrule where you first handled
the brackets then division then
multiplication in an addition and then
subtraction now in this case when you
see the denominator it is like a bracket
so you need to first sort out the
denominator part so what about the
denominator you have to same bases so
obviously the product would be also the
same base that is 3 what about the
powers the powers will get added because
here it is multiplication so the power
would get added so this is 3 to the
power 7 divided by 3 to the power 7 so
now again you have two numbers with the
same base so that means the product will
again have the same base 3 but what do
you do with the powers powers will get
subtracted because in this case it is
division so 7 minus 7 so 7 minus 7 is 0
so this is 3 to the power 0 anything to
the power 0 any integer to the power 0
is equal to 1 so therefore the result
would be 1 thank you please visit exam
fee or come for free quality education
you can learn with a simple four-step
learning process wherein you can watch
video lessons you can ask your questions
you can refer notes and you can take a
free online test we have content for
class 6 to 12 on physics chemistry
mathematics and biology along with
practical videos so please subscribe to
our channel for daily updates thank you
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Nov 08, 2024 Last updated |
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