Solving equations with radicals and exponents:
Exponents:
The term exponent in math is used to find the exponential value of the particular value it may be integer or fraction . We can easy to get the power of a number using online calculator ,Consider the unknown number , Here x is base and y is the power of x . Let us consider the number 23 Here 2 is the base(x) and 3 is the power of 2(y). We calculate 23= 2 x 2 x 2 =8
Radicals:
Square root of the number is said to be a radical number .A radical equation is an equation in which a variable is under a radical term.Let us discuss about the solving an equations with radical and exponents,
Example Problems to Solving Equations with Radicals and Exponents:
Example 1: Solve `sqrt(3x^2 +8x)` -3 =0
Solution:
Given `sqrt(3x^2+8x)` -3=0
isolate the radical equation,
`sqrt(3x^2 +8x)=3`
Take square on both sides we get,
`(sqrt3x^2+8x)^2` `=3^2`
3x^2+8x=9
3x^2+8x-9 =0
now you can solve the above equation using factoring method,
Using quadratic formula ,
x= `(-b+-sqrt(b^2-4ac))/(2a)`
Here b=8 ,a=3 and c=-9
Therefore ,
x= `(-8+-sqrt(8^2-(4)(3)(-9)))/(2(3))`
`=(-8+-sqrt(64+108))/6`
`=(-8+-sqrt(172))/6`
`=(-8+-sqrt(2*2*43))/6`
`=(-8+-2sqrt(43))/6`
`=2([-4+-sqrt(43)])/6`
`=(-4+-sqrt43)/3`
Therefore,
The factors are , `(-4+sqrt43)/3` ,`(-4-sqrt43)/3`
Example 2 : Solve `sqrt( x^2 +5x)-3=0`
Solution:
Isolate the given radical and exponent equation, we get,
`sqrt(x^2+5x)=3`
Take square root on both sides we get,
x^2+5x=32
x^2+5x=9
x^2+5x-9=0
Above equation in the form of ax^2+bx+c.
Therefore you can find the factors using quadratic formula,
`x= (-b+-sqrt(b^2-4ac))/(2a)`
Here , b=5,a=1 and c=-9
substitute these values into formula,
`x= ((-5+-sqrt(5^2-4(1)(-9)))/(2(1)))`
`=(-5+-sqrt(25+36))/2`
`=(-5+-sqrt(61))/2`
`=(-5+-sqrt(61))/2`
Therefore ,
The factors are ,
`x= (-5+sqrt61)/2` , `(-5-sqrt61)/2`
More about the Solving of Equations with Radicals and Exponents:
Example3: Solve `sqrt(2x^2+4)=4`
Solution:
Take square on both sides we get,
`(sqrt(2x^2+4))^2` =42
2x^2+4 =16
subtract both sides by 4,we get
2x^2+4-4=16-4
2x^2=12
Divide both sides by 2 ,
x^2 =`12/2`
x^2=6
x=`sqrt6`
Example 4: Solve the equations with radicals and exponents `sqrt(4x^2+8)=11`
Solution:
Take square on both sides , we get,
4x^2+8 =121
Subtract both sides by 8 we get,
4x^2+8-8 =121-8
4x^2=113
Divide both side by 4,
x^2 =113/4
x=`sqrt(113/4)`
x=`(sqrt113)/2`
Therefore the value of `x= sqrt113/2`
Example 5: Solve the equation with radicals and exponents `sqrt(6x^2+7)` `=8`
Solution:
Take square on both sides we get,
6x^2 +7 = 64
Subtract both side by7,
6x^2+7-7=64-7
6x^2=57
Divide both side by 6,
x^2 =`57/6`
x= `sqrt(57/6)`
Therefore the value of `x = sqrt(57/6)`
Exponents:
The term exponent in math is used to find the exponential value of the particular value it may be integer or fraction . We can easy to get the power of a number using online calculator ,Consider the unknown number , Here x is base and y is the power of x . Let us consider the number 23 Here 2 is the base(x) and 3 is the power of 2(y). We calculate 23= 2 x 2 x 2 =8
Radicals:
Square root of the number is said to be a radical number .A radical equation is an equation in which a variable is under a radical term.Let us discuss about the solving an equations with radical and exponents,
Example Problems to Solving Equations with Radicals and Exponents:
Example 1: Solve `sqrt(3x^2 +8x)` -3 =0
Solution:
Given `sqrt(3x^2+8x)` -3=0
isolate the radical equation,
`sqrt(3x^2 +8x)=3`
Take square on both sides we get,
`(sqrt3x^2+8x)^2` `=3^2`
3x^2+8x=9
3x^2+8x-9 =0
now you can solve the above equation using factoring method,
Using quadratic formula ,
x= `(-b+-sqrt(b^2-4ac))/(2a)`
Here b=8 ,a=3 and c=-9
Therefore ,
x= `(-8+-sqrt(8^2-(4)(3)(-9)))/(2(3))`
`=(-8+-sqrt(64+108))/6`
`=(-8+-sqrt(172))/6`
`=(-8+-sqrt(2*2*43))/6`
`=(-8+-2sqrt(43))/6`
`=2([-4+-sqrt(43)])/6`
`=(-4+-sqrt43)/3`
Therefore,
The factors are , `(-4+sqrt43)/3` ,`(-4-sqrt43)/3`
Example 2 : Solve `sqrt( x^2 +5x)-3=0`
Solution:
Isolate the given radical and exponent equation, we get,
`sqrt(x^2+5x)=3`
Take square root on both sides we get,
x^2+5x=32
x^2+5x=9
x^2+5x-9=0
Above equation in the form of ax^2+bx+c.
Therefore you can find the factors using quadratic formula,
`x= (-b+-sqrt(b^2-4ac))/(2a)`
Here , b=5,a=1 and c=-9
substitute these values into formula,
`x= ((-5+-sqrt(5^2-4(1)(-9)))/(2(1)))`
`=(-5+-sqrt(25+36))/2`
`=(-5+-sqrt(61))/2`
`=(-5+-sqrt(61))/2`
Therefore ,
The factors are ,
`x= (-5+sqrt61)/2` , `(-5-sqrt61)/2`
More about the Solving of Equations with Radicals and Exponents:
Example3: Solve `sqrt(2x^2+4)=4`
Solution:
Take square on both sides we get,
`(sqrt(2x^2+4))^2` =42
2x^2+4 =16
subtract both sides by 4,we get
2x^2+4-4=16-4
2x^2=12
Divide both sides by 2 ,
x^2 =`12/2`
x^2=6
x=`sqrt6`
Example 4: Solve the equations with radicals and exponents `sqrt(4x^2+8)=11`
Solution:
Take square on both sides , we get,
4x^2+8 =121
Subtract both sides by 8 we get,
4x^2+8-8 =121-8
4x^2=113
Divide both side by 4,
x^2 =113/4
x=`sqrt(113/4)`
x=`(sqrt113)/2`
Therefore the value of `x= sqrt113/2`
Example 5: Solve the equation with radicals and exponents `sqrt(6x^2+7)` `=8`
Solution:
Take square on both sides we get,
6x^2 +7 = 64
Subtract both side by7,
6x^2+7-7=64-7
6x^2=57
Divide both side by 6,
x^2 =`57/6`
x= `sqrt(57/6)`
Therefore the value of `x = sqrt(57/6)`
