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Peace be upon you, and Allah's mercy and

blessings. Back again on the

Kimatika channel. In this video, we will

learn about 10th grade mathematics material, namely

exponents and logarithms. In

this section, we will discuss

what exponents are and the properties of

exponents. Exponents are also called

powers. So, for example, if a number is known to have a

power of a^n, then n is what is

called the exponent or

power. Meanwhile, a is a number

called the base or

principal number. So, what is a

power number? Power numbers are a

form of repeated multiplication of a number

that we mentioned earlier by a

base number as many as its power or as many as

n times. So if we assume

the exponent is a^n, then

this means the same as a * a * a and

so on as many as n * or n factors.

Okay, now let's look at an example. The

first example is 3^4. This

means repeated multiplication of the number 3

to its power, namely 4 times.

That means it's the same as 3 * 3 * 3 * 3 = 81.

Clear, right? Let's look at the second example.

For example -5

with 3. Well, this means

repeated multiplication of the base number, namely -5.

as many as its power, namely 3 times.

It means the same as -5 * -5 * -5 or the

same as -125.

Okay, is that clear? Now we continue

discussing the properties of exponents. The

first property is that if there are two

exponents multiplied where the

base number is the same, then this

can be simplified to a^m + n.

So, if multiplied then the exponents are

added. Remember, the condition is that the

base numbers must be the same. So, the a's

have to be the same, right? Let's see

an example right away. For example 2^2 * 2^3. Well, this is

the same as 2^ the

sum of the powers added together, namely 2 +

3 or the same as 2^5. Continue with the

second property. If there is a

power number, for example a^ m divided by

another power number, for example a^ n

where the base number is the same, then

this can be simplified to a^ m -

n. So, his rank was reduced. Let's

see an example. For example 5^7 / 5^4. Well,

this is the same as 5^

7 - 4 or = 5^3. Clear, right? Carry on. The

third characteristic. If there is a number

with a power, for example a^ m, then

raise it to the power again by n. Well, this is the

same as

a^

m * n. So, the exponents are multiplied.

For example, suppose 3^2 is raised

to the power of 3. This means it is the same as 3^2 *

3 or 3^6. Next is the

fourth property. If two numbers are multiplied

and then raised to the power of m,

these numbers can be ordinary numbers or

exponents. Then

use brackets raised to the power of m, then

this is the same as the

first number, namely a raised

to the power of m and the second number, namely b

raised to the power of m. For example, 2^2

* 3^3. Now, this is raised to the power of 2.

This means that each number in the brackets is

raised to the power of 2 again

to become 2^2. The first number is

raised to the power of 2

times

3^3, the second number is raised

to the power of 2. Now, this can use the

previous property so that it becomes 2^

2 * 2 or 4 * 3^3 * 2, which is 6. Now,

can we simplify this? It's no

longer possible, because the base numbers are

different. Continuing the fifth characteristic.

If a is divided by b then

raised to the power of m, use brackets.

Well, this means the same as a raised to the power

of m and then divided by b raised to the power of

m. For example, 3/5

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raised to the power of 3 is the same as

3 raised to the power of 3 divided by 5

also raised to the power of 3. Okay, that's clear

. Next is the sixth characteristic. If

there is a number raised to the power of 0,

for example a^ 0, then this must be equal to

1. The condition is that the base number or a

cannot be equal to 0, yes. Example 3^

0 means this is equal to 1, yes.

Likewise, 5^ 0 is also the same

as 1. This means that whatever the number is

except 0, if it is raised to the power of 0,

the result is the same as 1. Next is the

seventh characteristic. If there is a

negative exponent number, for example

a^-m,

then we can change this into a

positive exponent number, which is equal to

1/a^

m.

For example, 3^-2, the

positive exponent form is

1/3^

2. Okay, let's continue. The

8th trait. If a number has a

fractional exponent, for example a^m/n, then

we can change it into a root form, yes. That is

√^ n of a^ m or vice versa, yes.

Example 3^

1/2 or 1/2. Well, this is the same as AK P

2. So, this is the n

of 3^

1. The M is 1. Well, for AK^ 2 you

don't need to write the 2. Then the 3^ 1^

1 also doesn't need to be written. This means that

3^1/2 is the same as √3.

Another example is 5^2/3. Well, this is the same

as √^3

of 5

^2. Okay, clear, right? So, there are eight

properties of exponents. To

understand it better, we will discuss some

example questions. First,

simplify 4x^ 7y^ 8 / 2 * x^ 3y^

2^

2. Okay, first let's pay attention,

which one should we

simplify first using the properties of

exponents that we discussed earlier. Well,

here we have x^3y^2^2. So this is what

we will analyze first.

The same as for the numerator,

we still write 4x^7 y^8 /

2 without including it in the brackets, meaning

we still write 2. So, this power of 2

does not belong to or only belongs to

the numbers in the

brackets. Then

we open the brackets of x^3, which means we

raise it to the power of 2

times y^2 raised to the power of 2

4 / 2. We can do this straight away,

the result is 2

= 2 x^7, we write it as y^8, we write it as y^8, we

divide it. Well, here we have x^3

raised to the power of 2 again, meaning

the exponent is multiplied to become

x^3 * 2 or 6. Then there is y^2

raised to the power of 2 again. The exponent is

also multiplied to become y^4. Okay,

now we continue using the property of

dividing exponents with the

same base number. This means

the exponent is reduced so that it becomes

equal to 2 * x^

7 - 6. So the exponent - x^7 / x^6

becomes x^7 - 6. Then for y^8 / y^4 it

becomes y^8 - 4 = 2 * x 7 - 1. There is no

need to write it for exponent 1. Then y^8

- 4 or 4. So the result of the simplification

of this problem is 2xy^

4. Clear, right? Let's move on to the second example question

. Simplify 5a² b^ 2 *

a^ 3b^ 4. Okay, let's simplify.

Likewise, note here that those

raised to the power of 2 are all the numbers

in brackets. This means it

includes the number or number 5.

This means 5^2 then a^2 is raised

to the power of 2. Then b^

2 di* a^3 b^4

= 5^2 is 25.

a^2 raised to the power of 2 means

the exponents are multiplied to become a^

4. Then there are b^2, a^3, b^4

= 25. This is a^4 * b^2 * a^3 * b^4.

This means there are the same base numbers,

namely a^4 * a^3, meaning the

exponents are added to become a^4 + 3.

Then there is b^2 * b^4. The exponents are also

added to become

a^4 + 3, namely 7 b^2 + 4, namely 6.

This means the result of the simplification for question

number 2 is 25a^7b^6. Is that clear?

Continue to the third example question.

Simplify it in

positive integer exponent form a^-3b^

7

/ a^ -5b^

4c^ 3. So this is division. To make it

easier to simplify, we change it

to fraction form so it

equals a^ -3b^

7 c.

divided or per a^ -5 b^ 4 c^ 3. Well,

we can immediately

simplify this using the exponent property of

division. So, if the base numbers are

the same, the exponents can be subtracted for

division. So it is the same as a^ -3 /

a^ -5 which means the exponent is reduced

to -3.

di-5

then multiplied by b^

7 / b^ 4 to the power of - 7 - 4 * c^

1 yes if not written

- 3

= a^ -3

- -5 this is negative meets negative becomes

positive yes that means negative 3 + 5 becomes

+2 or a^ 2 * b - 4 3 * c 1 - 2 1 - 3

is -2. Well, pay attention to the question

that is asked for simplification

into positive integer exponents.

Meanwhile, here there are still those with

negative powers. This means we can

change it to a positive exponent according

to the seventh property of exponents

. So it's the same as a^ 2 b^ 3 c^

-2 moves down to c^ 2.

Everything is positive. This means that the

simplification result for question number 3 is

a^ 2b^ 3/c^

2. Okay, let's move on to the fourth example question

.

Simplify √^ 3 from 8a^ 6b^ 2.

Now, remember one of the properties of exponents

is that the power form or

exponent form can be changed into the

root form. This means the opposite, the root form

can be changed into the exponent form. So it's the

same as being able to see the 8th property,

namely 8a a^ 6b^2

[Music]

raised to the power of 1/3.

So the power of 1/3 of √^ 3 is the

same as 8. We change it to a

power number form of 2^ 3 a^ 6 b^

2 1/3

then each number in

these brackets is raised to the power of 1/3. Well,

a number raised to a power

means that the power is multiplied, right? So

let's just multiply the exponents

for each number. So 2^

3 * 1/3 or 3/3

di* A^

6 * 1/3 or 6/3

* B^ 2 * 1/3 or 2/3

= 2^ 3/3 is 2^ 1^ 1 does not need to be

written multiplied by a 6/3^

6/3 means the power of

2 * b^ 2/3.

This means that the simplification result of question

number 4 is 2a^ 2b^ 2/3. Next is the

fifth example question. Determine the value of

x in the following equation. Well, let's

finish A first, okay? 3^ x =

27. To determine the value of x,

we change 27 into a

power number with the same base number

as the one on the left side, namely 3.

So it becomes 3^ x. Well, 27 is

3^

3.

The base numbers are the same, meaning the

exponents are also the same. means x = 3.

Then for B there is the equation a^ x *

a^ 4 = a^ 7. It is all in the form of

exponents and the base number is

also the same. For those on the left side,

multiplying means adding the exponents

so that it becomes a^ x + 4 = a^ 7. Now,

pay attention, the base numbers are the same,

OK?

This means the exponents are also the same, namely x +

4 = 7. Then we determine the value of x =

7. The

+4 moves to the right side to become -4.

Then x = 7 - 4 which is 3. So the value of x

for question a is 3. The value of x

for question b.

Okay, that's understandable. Well, that's all

for this video. Thank You.

Peace be upon you and Allah be upon you

.