Exponents and Powers Formulas

Can we read 10,000,000,000,000,000? These massive natural numbers are not easy to read as well as to recognize and hence evaluate. Exponents make it easy to read and handle very large numbers. We also call exponents, powers or indices in mathematics. Exponents and powers make the complex computations easy and faster. In this topic, we will discuss various exponents and powers formulas with examples. Also, the laws of exponents will be discussed. Let us learn it!

Exponents and Powers Formulas

What are Exponents and Powers?

We know how to calculate the expression like 6 × 6. This expression can be written in a shorter way using the exponents.So, \(6 \times 6 = 6 ^ 2\). Also, \(5 \times 4 = 5 ^ 4\)

An expression that represents repeated multiplication of the same factor is power. The number 6 is the base, and the number 2 is its exponent. The exponent corresponds to the number of times the base will be multiplied by itself.

Therefore, if two powers have the same base then we can multiply these two powers. When we multiply two powers, we will add their exponents. If two powers have the same base then we can divide the powers also. When we divide these powers then we subtract their exponents. Here in this topic, the student will learn about various formulas of exponents.

When we need to repeatedly multiply some number by itself, then it means that we raise it some power. We know this as the Exponent. The power in the exponent represents the number of times that we want to carry out the multiplication operation. Exponents are having their own set of rules when it comes to carrying out Arithmetic Operations. These formulas will help a lot with many mathematical simplifications.

Source:en.wikipedia.org

The Exponents and Powers Formulas are

Laws of exponents are used to simplify the calculations. Some important laws of exponents are:

1. Power zero \(a^{ 0 } = 1\)
2. Power one \(a^{ 1 } = a\)
3. Fraction formula \(\sqrt{ a } = a^{ \frac{ 1 }{ 2 }}\)
4. Reverse formula : \(\sqrt{ n } { a } = a^{\frac{ 1 } { n }}\)
5. Negative power value : \(a^{ -n } = \frac{ 1 }{ a^{n} }\)
6. Fraction formula : \(a^{n} = \frac{1}{ a^{ -n } }\)
7. Product formula : \(a^{m}a^{n} = a^{ m + n }\)
8. Division Formula : \(\frac{ a^{ m }}{ a^{ n }} = a ^{ m-n }\)
9. Power of Power formula : \((a^{ m })^{ p } = a^{ mp }\)
10. Power distribution Formula : \((a^ { m }c^{ n })^{ x } = a ^ { mx } c ^{ nx }\)
11. The Power distribution Formula : \(\left ( \frac {a ^{ m }}{c^{ n }} \right )^{x} = \frac{a^{ mx }}{c^{ nx }}\)

Solved Examples for Exponents and Powers Formulas

Example-1: Solve \({ 4 }^ { -3 }\)

Solution: As per the the negative exponent rule:

\(\large a^{ -n }=\frac{ 1 }{ a^{n} }\)

i.e. \(\large{ 4 } ^{ -3 }= \frac{ 1 }{ 4 ^ {3} }\)

i.e. \(\large { 4 } ^ {-3} = \frac { 1 }{ 64 }\)

Example-2:  Simplify : \({(x^3 y^4)} ^ 2\)

Solution: We will do its simplification as:

\((x^{ 3 }\cdot y^{ 4 })^{ 2 } = x^{ 3\cdot 2 }\cdot y^{ 4\cdot 2 }=x^{6}\cdot y^{ 8 }\)

Example-3: Simplify: \((\frac{4}{5})^5 \times (\frac{5}{6}) ^ 5\)

Solution : We can simplify it by using the laws of exponents.

\((\frac{ 4 }{ 5 })^5 \times (\frac{ 5 }{ 6 }) ^ 5 \\\)

\(= (\frac{ 4 }{ 5 } \times \frac { 5 }{ 6 }) ^ 5 = (\frac{ 4 }{ 6 }) ^5 \)

\(= (\frac{ 2 } { 3 }) ^5 \)