Algebra power of exponents
Introduction :
To find exponents in ma thematic are an efficient way to express repeated multiplications of the same number. Specifically powers of 10 express very large and very small numbers in an economical manner. When learn exponents online, the exponential notation is combined with decimal notation, very small and very large numbers can be written efficiently. Exponential notation is much more efficient for conveying numeric or quantitative information.
Let a be a positive real number, a `!=` 0 and the algebra power of exponent equation
y = ax
In the above equation to easily identify the parts of exponents. such as
1. the algebra power of exponents is 'y'.
2. the power is 'x'.
3. the base is 'a'.
Laws for algebra power of exponents:
In the algebra power of exponents supports the following laws, such as
1. xn = x . x . x . x . . . . x (n factors of x)
2. x-n = ( 1 / xn ) if x `!=` 0
3. x`(1)/(n)` = `root(n)(x)`
4. xa . xb = xa+b
5. xa . ya = (x . y)a
6. `(x^a)/(x^b)` = xa . x-b = xa-b
7. `((x^a)/(y^a))` = `((x)/(y))^a`
8. (xa)b = xa . b
9. x`(p)/(q)` = `root(q)(x^p)` = `(root(q)(x))^p`
10. x0 = 1
Examples for algebra power of exponents:
Example 1:
(a). Evaluate `x^2. x^a`
(b). Evaluate `4.16.64`
Solution:
(a). Given:
`x^2.x^a`
To apply the law an. am = an+m (same base 'a' and different power):
`(x^2).(x^a) = x^(2 + a)`
Therefore, the required algebra power of exponents of `(x^2)(x^a)` is `x^(2 + a)` .
(b). Given:
`4.16.64`
To apply the law an.bn = (ab)n (different base and same powers):
`(4)(16)(64) = (2^2)(4^2)(8^2)`
`= [(2)(4)(8)]^2`
`= (64)^2`
`= 4096.`
Therefore, the required algebra power of exponents of `(4)(16)(64)` is` 4096` or `64^2` .
Example 2: algebra power of exponents.
(a). Simplify `-82`
(b). Simplify `(a^2)^4` .
Solution:
(a). Given:
-`8^2`
To simplify:
`-8^2 = (-1)(8)^2`
`= (-1)(8)(8)`
`= (-1)(64)`
`= -64` .
Therefore, the required algebra power of exponents of `-8^2` is `-64` .
(b). Given:
`(a^2)^4`
To simplify:
`(a^2)^4 = (a^2)(a^2)(a^2)(a^2)`
`= (a . a)(a . a)(a . a)(a . a)`
`= (a^1 . a^1)(a^1 . a^1)(a^1 . a^1)(a^1 . a^1)`
`= a^(1 + 1 + 1 + 1+ 1 + 1+ 1 + 1)`
`= a^8`
Therefore, the required algebra power of exponents of `(a^2)^4` is `a^8` .
I like to share this Law of Exponents and Simplify Exponents with you all through my blog.
To find exponents in ma thematic are an efficient way to express repeated multiplications of the same number. Specifically powers of 10 express very large and very small numbers in an economical manner. When learn exponents online, the exponential notation is combined with decimal notation, very small and very large numbers can be written efficiently. Exponential notation is much more efficient for conveying numeric or quantitative information.
Let a be a positive real number, a `!=` 0 and the algebra power of exponent equation
y = ax
In the above equation to easily identify the parts of exponents. such as
1. the algebra power of exponents is 'y'.
2. the power is 'x'.
3. the base is 'a'.
Laws for algebra power of exponents:
In the algebra power of exponents supports the following laws, such as
1. xn = x . x . x . x . . . . x (n factors of x)
2. x-n = ( 1 / xn ) if x `!=` 0
3. x`(1)/(n)` = `root(n)(x)`
4. xa . xb = xa+b
5. xa . ya = (x . y)a
6. `(x^a)/(x^b)` = xa . x-b = xa-b
7. `((x^a)/(y^a))` = `((x)/(y))^a`
8. (xa)b = xa . b
9. x`(p)/(q)` = `root(q)(x^p)` = `(root(q)(x))^p`
10. x0 = 1
Examples for algebra power of exponents:
Example 1:
(a). Evaluate `x^2. x^a`
(b). Evaluate `4.16.64`
Solution:
(a). Given:
`x^2.x^a`
To apply the law an. am = an+m (same base 'a' and different power):
`(x^2).(x^a) = x^(2 + a)`
Therefore, the required algebra power of exponents of `(x^2)(x^a)` is `x^(2 + a)` .
(b). Given:
`4.16.64`
To apply the law an.bn = (ab)n (different base and same powers):
`(4)(16)(64) = (2^2)(4^2)(8^2)`
`= [(2)(4)(8)]^2`
`= (64)^2`
`= 4096.`
Therefore, the required algebra power of exponents of `(4)(16)(64)` is` 4096` or `64^2` .
Example 2: algebra power of exponents.
(a). Simplify `-82`
(b). Simplify `(a^2)^4` .
Solution:
(a). Given:
-`8^2`
To simplify:
`-8^2 = (-1)(8)^2`
`= (-1)(8)(8)`
`= (-1)(64)`
`= -64` .
Therefore, the required algebra power of exponents of `-8^2` is `-64` .
(b). Given:
`(a^2)^4`
To simplify:
`(a^2)^4 = (a^2)(a^2)(a^2)(a^2)`
`= (a . a)(a . a)(a . a)(a . a)`
`= (a^1 . a^1)(a^1 . a^1)(a^1 . a^1)(a^1 . a^1)`
`= a^(1 + 1 + 1 + 1+ 1 + 1+ 1 + 1)`
`= a^8`
Therefore, the required algebra power of exponents of `(a^2)^4` is `a^8` .
I like to share this Law of Exponents and Simplify Exponents with you all through my blog.