10 To The Fourth Power
In mathematics, the exponents and powers terms are employed when a number is multiplied by itself by a certain number of times. For example, 4 × 4 × four= 64. This can also be written in curt form as 43 = 64. Here, four3 means the number "4" is multiplied by itself by 3 times, and brusk-grade 4three is the exponential expression. The number "iv" is the base number, while the number "three" is the exponent, and we read the given exponential expression every bit "4 raised to the power of 3". In an exponential expression, the base is the gene that is multiplied repeatedly by itself, whereas the exponent is the number of times the factor appears.
Definition of Exponents and powers
If a number is multiplied by itself n times, the resulting expression is known as the nth ability of the given number. There is a very thin line of difference between exponent and ability. An exponent is the number of times a given number has been multiplied by itself, while the power is the value of the production of the base number raised to an exponent. With the help of the exponential form of numbers, we tin more conveniently express extremely big and minor numbers. For instance, 100000000 can be expressed as 1 × 108, and 0.0000000000013 can exist expressed every bit thirteen × 10-13. This makes numbers easier to read, aids in maintaining their accuracy, and likewise saves us time.
Rules of Exponents and Powers
The rules of exponents and powers explain how to add together, subtract, multiply, and divide exponents as well as how to solve various kinds of mathematical equations involving exponents and powers.
| Product Law of Exponents | ayard × anorthward=a(m+ n) |
|---|---|
| Quotient Rule of Exponents | ayard/anorthward=a(chiliad-due north) |
| Power of a power rule | (am)n = amn |
| Power of a product dominion | athousand × bm = (ab)chiliad |
| Power of a quotient rule | am/bone thousand = (a/b)m |
| Nothing Exponent rule | a0 = 1 |
| Negative Exponent Rule | a-m = 1/am |
| Partial Exponent Rule | a(thou/northward) = due north√a1000 |
Dominion 1: Product Law of Exponents
According to this law, when exponents with the aforementioned bases are multiplied, the exponents are added together.
Product Law of Exponents: am × an=a(k+ n)
Rule 2: Caliber Dominion of Exponents
According to this constabulary, to divide ii exponents with the same bases, we demand to decrease the exponents.
Caliber Dominion of Exponents: athousand/anorthward=a(g–north)
Rule 3: Power of a power rule
According to this law, if an exponential number is raised to some other power, then the powers are multiplied.
Power of a power rule: (am)n=a(m× n)
Rule four: Power of a product dominion
According to this law, we need to multiply the different bases and raise the same exponent to the product of bases.
Ability of a product rule: am × bm=(a × b)thou.
Rule 5: Power of a quotient rule
According to this law, we need to carve up the unlike bases and raise the same exponent to the quotient of bases.
Power of a quotient dominion: ayard ÷ bm=(a/b)m
Rule 6: Cypher Exponent dominion
According to this law, if the value of a base of operations raised to the ability of goose egg is 1.
Aught Exponent rule: a0=i
Rule 7: Negative Exponent Rule
Co-ordinate to this law, if an exponent is negative, then irresolute the exponent to positive by taking the reciprocal of an exponential number.
Negative Exponent Dominion: a-m = 1/am
Rule 8: Partial Exponent Rule
Co-ordinate to this law, when we have a partial exponent, and so it results in radicals.
Fractional Exponent Rule: a(1/northward) = north√a
a(grand/northward) = n√am
What does 10 to the ability of iv hateful?
Solution:
Let usa calculate the value of 10 to the 4th mean, i.e., 10four
Nosotros know that according to the power rule of exponents,
am = a × a × a… k times
Hence, nosotros tin can write 104 as 10 × x × 10 × 10 = 10000
Therefore,
the value of 10 raised to the power of four, i.e., 10four is 10000.
Sample Issues
Problem i: Detect the value of 36.
Solution:
The given expression is iii6.
The base of the given exponential expression is "iii", while the exponent is "six", i.east., the given expression is read as "three is raised to the power of 6."
So, by expanding 36, nosotros go 3vi = 3 × iii × 3 × 3 × 3 × 3 = 729
Hence, the value of 3half-dozen is 729.
Problem two: Determine the exponent and power for the expression (12)5.
Solution:
The given expression is 12v.
The base of the given exponential expression is "12", while the exponent is "v", i.e., the given expression is read as "12 is raised to the power of 5."
Problem 3: Evaluate (2/7)–five × (two/7)7.
Solution:
Given: (two/seven)–5×(2/7)7
Nosotros know that, agrand × an = a(thou + due north)
So, (ii/7)–v×(ii/7)seven = (ii/7)(-5+7)
= (ii/7)2 = 4/49
Hence, (2/seven)–5 × (2/7)7 = 4/49
Problem 4: Discover the value of 10 in the given expression: v3x-2 = 625.
Solution:
Given, 53x-two = 625.
53x-2 = 54
Past comparing the exponents of the similar base of operations, nosotros get
⇒ 3x -2 = 4
⇒ 3x = four + two = 6
⇒ x = 6/3 = 2
Hence, the value of ten is 2.
Problem 5: Discover the value of k in the given expression: (-2/3)4× (2/3)-15 = (ii/3)7k+three
Solution:
Given,
(-2/iii)4× (2/three)-xv = (2/iii)7k+three
⇒ (2/3)4× (2/3)-15 = (2/3)7k+iii {Since (-ten)4 = ten4}
We know that, achiliad × an = a(m + n)
⇒ (ii/3)iv-15 = (2/three)7k+3
⇒ (ii/3)-11 = (2/iii)7k+three
Past comparing the exponents of the similar base, nosotros get
⇒ -xi = 7k +iii
⇒ 7k = -xi-three = -14
⇒ one thousand = -xiv/vii = -2
Hence, the value of g is -2.
10 To The Fourth Power,
Source: https://www.geeksforgeeks.org/what-is-10-to-the-4th-power/#:~:text=the%20value%20of%2010%20raised,i.e.%2C%20104%20is%2010000.
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