What is 10 to the 4th 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 × 4= 64. This can also be written in short form as 4³ = 64. Here, 4³ means the number “4” is multiplied by itself by three times, and short-form 4³ is the exponential expression. The number “4” is the base number, while the number “3” is the exponent, and we read the given exponential expression as “4 raised to the power of 3”. In an exponential expression, the base is the factor 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 power of the given number. There is a very thin line of difference between exponent and power. An exponent is the number of times a given number has been multiplied by itself, while the power is the value of the product of the base number raised to an exponent. With the help of the exponential form of numbers, we can more conveniently express extremely large and small numbers. For instance, 100000000 can be expressed as 1 × 10⁸, and 0.0000000000013 can be expressed as 13 × 10^-13. This makes numbers easier to read, aids in maintaining their accuracy, and also saves us time. 

Rules of Exponents and Powers

The rules of exponents and powers explain how to add, subtract, multiply, and divide exponents as well as how to solve various kinds of mathematical equations involving exponents and powers.

Product Law of Exponents	am × an=a(m+ n)
Quotient Rule of Exponents	am/an=a(m-n)
Power of a power rule	(am)n = amn
Power of a product rule	am × bm = (ab)m
Power of a quotient rule	am/bm = (a/b)m
Zero Exponent rule	a0 = 1
Negative Exponent Rule	a-m = 1/am
Fractional Exponent Rule	a(m/n) = n√am

Rule 1: Product Law of Exponents

According to this law, when exponents with the same bases are multiplied, the exponents are added together.

Product Law of Exponents: a^m × aⁿ=a^{(m+ n)}

Rule 2: Quotient Rule of Exponents

According to this law, to divide two exponents with the same bases, we need to subtract the exponents.

Quotient Rule of Exponents: a^m/aⁿ=a^(m–n)

Rule 3: Power of a power rule

According to this law, if an exponential number is raised to another power, then the powers are multiplied.

Power of a power rule: (a^m)ⁿ=a^{(m× n)}

Rule 4: Power of a product rule

According to this law, we need to multiply the different bases and raise the same exponent to the product of bases.

Power of a product rule: a^m × b^m=(a × b)^m.

Rule 5: Power of a quotient rule

According to this law, we need to divide the different bases and raise the same exponent to the quotient of bases.

Power of a quotient rule: a^m ÷ b^m=(a/b)^m

Rule 6: Zero Exponent rule

According to this law, if the value of a base raised to the power of zero is 1.

Zero Exponent rule: a⁰=1

Rule 7: Negative Exponent Rule

According to this law, if an exponent is negative, then changing the exponent to positive by taking the reciprocal of an exponential number.

Negative Exponent Rule: a^-m = 1/a^m

Rule 8: Fractional Exponent Rule

According to this law, when we have a fractional exponent, then it results in radicals.

Fractional Exponent Rule: a^(1/n) = ⁿ√a

a^(m/n) = ⁿ√a^m

What does 10 to the power of 4 mean?

Solution:

Let us calculate the value of 10 to the 4th mean, i.e., 10⁴

We know that according to the power rule of exponents,

a^m = a × a × a… m times

Hence, we can write 10⁴ as 10 × 10 × 10 × 10 = 10000

Therefore, 

the value of 10 raised to the power of 4, i.e., 10⁴ is 10000.

Sample Problems

Problem 1: Find the value of 3⁶.

Solution:

The given expression is 3⁶.

The base of the given exponential expression is “3”, while the exponent is “6”, i.e., the given expression is read as “3 is raised to the power of 6.”

So, by expanding 3⁶, we get 3⁶ = 3 × 3 × 3 × 3 × 3 × 3 = 729

Hence, the value of 3⁶ is 729.

Problem 2: Determine the exponent and power for the expression (12)⁵.

Solution:

The given expression is 12⁵.

The base of the given exponential expression is “12”, while the exponent is “5”, i.e., the given expression is read as “12 is raised to the power of 5.”

Problem 3: Evaluate (2/7)^–5 × (2/7)⁷.

Solution:

Given: (2/7)^–5×(2/7)⁷

We know that, a^m × aⁿ = a^{(m + n)}

So, (2/7)^–5×(2/7)⁷ = (2/7)^(-5+7)

= (2/7)² = 4/49

Hence, (2/7)^–5 × (2/7)⁷ = 4/49

Problem 4: Find the value of x in the given expression: 5^3x-2 = 625.

Solution:

Given, 5^3x-2 = 625.

5^3x-2 = 5⁴

By comparing the exponents of the similar base, we get

⇒ 3x -2 = 4

⇒ 3x = 4 + 2 = 6

⇒ x = 6/3 = 2

Hence, the value of x is 2.

Problem 5: Find the value of k in the given expression: (-2/3)⁴× (2/3)^-15 = (2/3)^7k+3

Solution:

Given,

(-2/3)⁴× (2/3)^-15 = (2/3)^7k+3

⇒ (2/3)⁴× (2/3)^-15 = (2/3)^7k+3  {Since (-x)⁴ = x⁴}

We know that, a^m × aⁿ = a^{(m + n)}

⇒ (2/3)^4-15 = (2/3)7k+3

⇒ (2/3)^-11 = (2/3)^7k+3

By comparing the exponents of the similar base, we get

⇒ -11 = 7k +3

⇒ 7k = -11-3 = -14

⇒ k = -14/7 = -2

Hence, the value of k is -2.