Solve using negative exponent
Introduction to exponents:
In algebra, there are many important topics. Exponents is one of the basic topics in algebra.
If x is a rational number and m is a positive integer, then `x^ m` `= x xx x xx x xx x xx . . .` m times.
If x is a non-zero rational number and k is a negative integral exponent, say k = – m, where m is a positive integer, then
`x^k = x^(-m)` `= x^(-1) xx x^(-1)xx . . .,` m times
`= (1/x)xx(1/x)....,` m times
`= (1/x)^m`
Solve using negative exponent:
In algebra, the negative rational exponents, If m is a positive integer and x is a non-zero rational number, then the negative rational exponent looks
`x^(-m) = 1/ (x^m) = (1/ x)^m ` ,
i.e., x– m is the reciprocal of xm or the mth power of the reciprocal of x. If p/q is a positive rational number and x > 0 is a rational number, then
`x^(-p/q) = 1/(x^(p/q)) = (1/x)^(p/q) ` ,
i.e., `x^(-p/q)` is the reciprocal of `x^(p/q)`
Ex1: Solve using negative exponent .
Solve using negative exponent: `5^(-3)` .
Solution:
Given:
`5^(-3)` .
To solve using negative exponent:
`5^(-3) =1/5^(3)` . [`5^(-3) ` is the reciprocal of `5^3` or the 3rd power of the reciprocal of 5]
`= (1/125)` . [Here, `5^3 = 5 xx 5 xx 5 = 25 xx 5 = 125`].
Therefore, the required solved negative exponent of `5^(-3)` is `1/125` .
Ex2: Solve using negative exponent.
Solve using negative exponent: `8^(-5/3)` .
Solution:
Given:
`8^(-5/3)` .
To solve using negative exponent:
`8^(-5/3) =1/8^(5/3)` . [ `8^(-5/3)` is the reciprocal of `8^(5/3)` or the `(5/3)^(th)` power of the reciprocal of 8 ]
`= (1/8)^(5/3)` .
`= [(1/8)^(1/3)]^5` . [Here, we are using `a^(p/q) = [a^(1/q)]^p` ].
`=[(1/2^3)^(1/3)]^5` [Here, we are using the expanding of 8 is `2 xx 2 xxx 2 = 2^3` ]
`=[1/2^(3xx(1/3))]^5`
`=[1/2]^5`
`=[1/2^5]`
`=[1/32]`
Therefore, the required solved negative exponent of `8^(-5/3)` is `1/32` .
Ex3: Solve using negative exponent.
Solve using negative exponent: `3^(-4/7) = 9^(4x)` .
Solution:
Given:
`3^(-4/7) = 9^(4x)` .
To solve using negative exponent:
`3^(-4/7) = 9^(4x)` .
`3^(-4/7) = (3^2)^(4x)` .
`3^(-4/7) = [3^(2 xx 4x)]` .
`3^(-4/7) = [3^(8x)]` .
`(-4/7) = [8x]` . Since, the given exponents of expression had the base values are equal.
`8x = -4/7` .
`(8x)/8 = (-4/7)/8` .
`(8x)/8 = -4/(7 xx 8)` .
`x = -1/(7 xx 2)` .
`x = -1/(14)` .
Therefore, the required solved negative exponents `x` value is `-1/14` .
In algebra, there are many important topics. Exponents is one of the basic topics in algebra.
If x is a rational number and m is a positive integer, then `x^ m` `= x xx x xx x xx x xx . . .` m times.
If x is a non-zero rational number and k is a negative integral exponent, say k = – m, where m is a positive integer, then
`x^k = x^(-m)` `= x^(-1) xx x^(-1)xx . . .,` m times
`= (1/x)xx(1/x)....,` m times
`= (1/x)^m`
Solve using negative exponent:
In algebra, the negative rational exponents, If m is a positive integer and x is a non-zero rational number, then the negative rational exponent looks
`x^(-m) = 1/ (x^m) = (1/ x)^m ` ,
i.e., x– m is the reciprocal of xm or the mth power of the reciprocal of x. If p/q is a positive rational number and x > 0 is a rational number, then
`x^(-p/q) = 1/(x^(p/q)) = (1/x)^(p/q) ` ,
i.e., `x^(-p/q)` is the reciprocal of `x^(p/q)`
Ex1: Solve using negative exponent .
Solve using negative exponent: `5^(-3)` .
Solution:
Given:
`5^(-3)` .
To solve using negative exponent:
`5^(-3) =1/5^(3)` . [`5^(-3) ` is the reciprocal of `5^3` or the 3rd power of the reciprocal of 5]
`= (1/125)` . [Here, `5^3 = 5 xx 5 xx 5 = 25 xx 5 = 125`].
Therefore, the required solved negative exponent of `5^(-3)` is `1/125` .
Ex2: Solve using negative exponent.
Solve using negative exponent: `8^(-5/3)` .
Solution:
Given:
`8^(-5/3)` .
To solve using negative exponent:
`8^(-5/3) =1/8^(5/3)` . [ `8^(-5/3)` is the reciprocal of `8^(5/3)` or the `(5/3)^(th)` power of the reciprocal of 8 ]
`= (1/8)^(5/3)` .
`= [(1/8)^(1/3)]^5` . [Here, we are using `a^(p/q) = [a^(1/q)]^p` ].
`=[(1/2^3)^(1/3)]^5` [Here, we are using the expanding of 8 is `2 xx 2 xxx 2 = 2^3` ]
`=[1/2^(3xx(1/3))]^5`
`=[1/2]^5`
`=[1/2^5]`
`=[1/32]`
Therefore, the required solved negative exponent of `8^(-5/3)` is `1/32` .
Ex3: Solve using negative exponent.
Solve using negative exponent: `3^(-4/7) = 9^(4x)` .
Solution:
Given:
`3^(-4/7) = 9^(4x)` .
To solve using negative exponent:
`3^(-4/7) = 9^(4x)` .
`3^(-4/7) = (3^2)^(4x)` .
`3^(-4/7) = [3^(2 xx 4x)]` .
`3^(-4/7) = [3^(8x)]` .
`(-4/7) = [8x]` . Since, the given exponents of expression had the base values are equal.
`8x = -4/7` .
`(8x)/8 = (-4/7)/8` .
`(8x)/8 = -4/(7 xx 8)` .
`x = -1/(7 xx 2)` .
`x = -1/(14)` .
Therefore, the required solved negative exponents `x` value is `-1/14` .