In mathematics, the terms “exponents and powers” are used when a number is multiplied by itself several times. A number is raised to the power of a natural number is equivalent to the number of times the number is multiplied by itself. When a number is multiplied by itself n times, the expression obtained is referred to as the nth power of the given number. The number that is multiplied repeatedly by itself is known as the base, and the number of times the number is being multiplied is known as the exponent.
For instance, 6 × 6 × 6 × 6 = 1296. Now, this is written as 64 in its exponential form, where 64 means the number “6” is multiplied by itself by four times. In the given exponential expression, “6” is the base number, and “3” is the exponent, and we read it as “6 raised to the power of 4”. The exponential form of numbers helps us in writing to express extremely large and small numbers more conveniently i.e. standard form or scientific form of number are expressed using exponents. For example, the speed of light in a vacuum is 29,97,92,458 m/s and this can be expressed approximately as 3 × 108 m/s and 0.000000000021 can be expressed as 21 × 10-11. This makes numbers easier to read, aids to maintain the accuracy of numbers, and also saves us time.
General rules of Exponents and Powers
The rules of exponents and powers, also known as the “law of exponents”, aid in solving the mathematical equations involving exponents and powers. With the help of the law of exponents, we can conveniently perform arithmetic operations such as addition, subtraction, multiplication, and division.
- Zero Exponent rule: This law states that, if the value of a base raised to the power of zero is 1.
a0=1
- Negative Exponent Rule: This law states that, if an exponent is negative, then changing the exponent to positive by taking the reciprocal of an exponential number.
a-n = 1/an
- Product Law of Exponents: This law states that, if the exponential expressions that have the same bases are multiplied, then add their exponents.
am × an = a(m+ n)
- Quotient Rule of Exponents: This law states that, if two exponential expressions that have the same bases are divided, then subtract their exponents.
am/an = a(m–n)
- Power of a power rule: This law states that if an exponential number is raised to another power, then the powers are multiplied.
(am)n = a(m× n)
- Power of a product rule: This law states that if the exponential expressions having different bases are multiplied and raise the same exponent to the product of bases.
am × bm = (a × b)m
- Power of a quotient rule: This law states that if the exponential expressions that have different bases are divided, then raise the same exponent to the quotient of bases.
am ÷ bm = (a/b)m
- Fractional Exponent Rule: This law states that, when the given exponential expression has a fractional exponent, then it results in radicals.
a(1/n) = n√a
a(m/n) = n√am
What is the power notation of 343?
Solution:
The given number is 343.
For writing the numbers in exponential form or power notation form, we need to express them raised to certain powers of their prime factors
Now, factorize the given number into its small factors.
Hence, the number 343 in terms of its factors is written as
343 = 7 × 7 × 7
Now, write it in the exponent form of power form. An exponent is the number of times the number is being multiplied by itself.
Here, the exponent is three as the number 7 is being multiplied by itself three times.
Hence, 343 in its power notation form/exponential form is written as,
343 =73
Therefore, 343 is equal to 73 or we can write it as “7 is raised to the power 3”.
Problems based on exponent and powers
Problem 1: Evaluate (5/8)–3 × (5/8)8.
Solution:
Given: (5/8)–3×(5/8)8
We know that, am × an = a(m + n)
So, (5/8)–3× (5/8)8 = (5/8)(-3+8)
= (5/8)5 = 3125/32,768
Hence, (5/8)–3 × (5/8)8 = 3125/32,768.
Problem 2: Find the value of x in the given expression: (3/5)5× (-3/5)-12 = (3/5)4x+5.
Solution:
Given,
(3/5)5× (-3/5)-12 = (3/5)4x+5
⇒ (3/5)5× (3/5)-12 = (3/5)4x+5 {Since (-x)-12 = x-12}
We know that, am × an = a(m + n)
⇒ (3/5)5-12 = (3/5)4x+5
⇒(3/5)-7 = (3/5)4x+5
By comparing the exponents of the similar base, we get
⇒ -7 = 4x+5
⇒ 4x = – 7 – 5 = -12
⇒ x = -12/4 = -3
Hence, the value of x is -3.
Problem 3: What is the exponent and power in the expression (7)13?
Solution:
The given expression is 713.
The base of the given exponential expression is “7”, while the exponent is “13”, i.e., the given expression is read as “7 is raised to the power of 13.”
Problem 4: What is the power notation of 14,641?
Solution:
The given number is 14,641.
On the prime factorization of 14641, we get 14641 = 11 × 11 × 11 × 11,
i.e., 14641 = 11 × 11 × 11 × 11 = 114
Hence, 14,641 is equal to 114 or we can write it as “11 is raised to the power 4”.
Problem 5: What does 5 to the power of 3 mean?
Solution:
Let us calculate the value of 5 to the power 3, i.e., 53.
According to the power rule of exponents,
am = a × a × a… m times
Hence, we can write 53 as 5 × 5 × 5 = 125
Therefore, the value of 5 raised to the power of 3, i.e., 53 is 125.
Problem 6: Find the value of k in the given expression: 42k-5 = 1024.
Solution:
Given, 42k-5 = 1024.
42k-5 = 45
By comparing the exponents of the similar base, we get
⇒ 2k – 5 = 5
⇒ 2k = 5 + 5 = 10
⇒ k = 10/2 = 5
Hence, the value of k is 5.