__maths__>__Foundation of Algebra with Numerical Arithmetics__>__Numerical Arithmetics: Laws and Properties for Algebra__### Laws and Properties of Arithmetics : Exponents

This page introduces the the properties of exponents. To understand polynomials and equations in algebra, these properties are used.

*click on the content to continue..*

Which of the following equals `p xx p xx p ... (m text( times))`?

- `p^m`
- `p^m`
- `m p`

The answer is `p^m`

Which of the following equals `p^m xx q^m`?

- `mab`
- `(p xx q)^m`
- `(p xx q)^m`

The answer is `(p xx q)^m`.

`p^m xx q^m`

`=p xx p xx ... (m text( times)) ``xx q xx q xx ... (m text( times))`

`=pq xx pq xx ... (m text( times)) `

`=(pq)^m`

Which of the following equals `p^m -: p^n`?

- `p^(m-n)`
- `p^(m-n)`
- `p^(m/n)`

The answer is `p^(m-n)`

Which of the following equals `p^m -: q^m`?

- `p/q`
- `(p -: q)^m`
- `(p -: q)^m`

The answer is `(p -: q)^m`

Which of the following equals `p^0`?

- `0`
- `1`
- `1`

The answer is `1`.

`p^0= p^(1-1) = p/p =1`

Which of the following equals `(p^m)^n`?

- `p^(m^n)`
- `p^(mn)`
- `p^(mn)`

The answer is `p^(mn)`

Which of the following equals `root(m)(p)`?

- `p^(1/m)`
- `p^(1/m)`
- `p^(-m)`

The answer is `p^(1/m)`

Which of the following equals `p^(-m)`?

- `1/(p^m)`
- `1/(p^m)`
- `-p^m`

The answer is `1/(p^m)`

Which of the following equals `p^m + p^m + …(n text ( times))`?

- `np^m`
- `np^m`
- `p^(mn)`

The answer is `np^m`

Which of the following equals `k xx p^m + l xx p^m`?

- `(k+l)xx p^m`
- `(k+l)xx p^m`
- `kl p^m`

The answer is `(k+l)xx p^m`

Which of the following equals `k xx p^m + l xx p^n`?

- `(k+l)xx p^m`
- it cannot be simplified
- it cannot be simplified

The answer is : it cannot be simplified. Expressions of this type lead to the definition of algebraic expressions and polynomials in the form `ax^m+bx^n+c`.

Which of the following equals `(p+q)^m`?

- `p^m+q^m`
- it cannot be simplified
- it cannot be simplified

The answer is : it cannot be simplified. Expressions of this type lead to the definition of algebraic identities.

**Properties of Exponents**

`a^m + a^m + ... (n text( times)) = n a^m`

`pa^m + qa^m = (p+q)a^m`

`a^m xx a^n = a^(m+n)`

`a^m xx b^m = (a xx b)^m`

`a^m -: a^n = a^(m-n)`

`a^m -: b^m = (a -: b)^m`

`(a^m)^n = a^(mn)`

`a^(1/m) = root(m)(a)`

`a^(-m) = 1/(a^m)`

`log_a a^m = m`

• Some forms cannot be simplified any further*That is, the expressions can be evaluated to equivalent numerical values, but not simplified retaining the power of `a`*

→ `pa^m + qa^n + r` cannot be simplified for `m != n` * This is the basis for algebraic expressions and polynomials *

→ `(a+b)^m` cannot be simplified, but an equivalent expression can be defined in the general form.*This is the basis for algebraic identities.*

*slide-show version coming soon*