# Solve Using the Square Root Property (( square root of 5)/2)^2=((3( square root of 2))/4)^2+x^2

(52)2=(3(2)4)2+x2
Rewrite the equation as (3(2)4)2+x2=(52)2.
(3(2)4)2+x2=(52)2
Simplify each term.
Use the power rule (ab)n=anbn to distribute the exponent.
Apply the product rule to 324.
(32)242+x2=(52)2
Apply the product rule to 32.
322242+x2=(52)2
322242+x2=(52)2
Simplify the numerator.
Raise 3 to the power of 2.
92242+x2=(52)2
Rewrite 22 as 2.
Use axn=axn to rewrite 2 as 212.
9(212)242+x2=(52)2
Apply the power rule and multiply exponents, (am)n=amn.
9⋅212⋅242+x2=(52)2
Combine 12 and 2.
9⋅22242+x2=(52)2
Cancel the common factor of 2.
Cancel the common factor.
9⋅22242+x2=(52)2
Divide 1 by 1.
9⋅2142+x2=(52)2
9⋅2142+x2=(52)2
Evaluate the exponent.
9⋅242+x2=(52)2
9⋅242+x2=(52)2
9⋅242+x2=(52)2
Raise 4 to the power of 2.
9⋅216+x2=(52)2
Multiply 9 by 2.
1816+x2=(52)2
Cancel the common factor of 18 and 16.
Factor 2 out of 18.
2(9)16+x2=(52)2
Cancel the common factors.
Factor 2 out of 16.
2⋅92⋅8+x2=(52)2
Cancel the common factor.
2⋅92⋅8+x2=(52)2
Rewrite the expression.
98+x2=(52)2
98+x2=(52)2
98+x2=(52)2
98+x2=(52)2
Simplify (52)2.
Apply the product rule to 52.
98+x2=5222
Rewrite 52 as 5.
Use axn=axn to rewrite 5 as 512.
98+x2=(512)222
Apply the power rule and multiply exponents, (am)n=amn.
98+x2=512⋅222
Combine 12 and 2.
98+x2=52222
Cancel the common factor of 2.
Cancel the common factor.
98+x2=52222
Divide 1 by 1.
98+x2=5122
98+x2=5122
Evaluate the exponent.
98+x2=522
98+x2=522
Raise 2 to the power of 2.
98+x2=54
98+x2=54
Move all terms not containing x to the right side of the equation.
Subtract 98 from both sides of the equation.
x2=54-98
To write 54 as a fraction with a common denominator, multiply by 22.
x2=54⋅22-98
Write each expression with a common denominator of 8, by multiplying each by an appropriate factor of 1.
Multiply 54 and 22.
x2=5⋅24⋅2-98
Multiply 4 by 2.
x2=5⋅28-98
x2=5⋅28-98
Combine the numerators over the common denominator.
x2=5⋅2-98
Simplify the numerator.
Multiply 5 by 2.
x2=10-98
Subtract 9 from 10.
x2=18
x2=18
x2=18
Take the square root of both sides of the equation to eliminate the exponent on the left side.
x=±18
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite 18 as 18.
x=±18
Any root of 1 is 1.
x=±18
Simplify the denominator.
Rewrite 8 as 22⋅2.
Factor 4 out of 8.
x=±14(2)
Rewrite 4 as 22.
x=±122⋅2
x=±122⋅2
Pull terms out from under the radical.
x=±122
x=±122
Multiply 122 by 22.
x=±122⋅22
Combine and simplify the denominator.
Multiply 122 and 22.
x=±2222
Move 2.
x=±22(22)
Raise 2 to the power of 1.
x=±22(22)
Raise 2 to the power of 1.
x=±22(22)
Use the power rule aman=am+n to combine exponents.
x=±2221+1
x=±2222
Rewrite 22 as 2.
Use axn=axn to rewrite 2 as 212.
x=±22(212)2
Apply the power rule and multiply exponents, (am)n=amn.
x=±22⋅212⋅2
Combine 12 and 2.
x=±22⋅222
Cancel the common factor of 2.
Cancel the common factor.
x=±22⋅222
Divide 1 by 1.
x=±22⋅2
x=±22⋅2
Evaluate the exponent.
x=±22⋅2
x=±22⋅2
x=±22⋅2
Multiply 2 by 2.
x=±24
x=±24
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the ± to find the first solution.
x=24
Next, use the negative value of the ± to find the second solution.
x=-24
The complete solution is the result of both the positive and negative portions of the solution.
x=24,-24
x=24,-24
x=24,-24
The result can be shown in multiple forms.
Exact Form:
x=24,-24
Decimal Form:
x=0.35355339…,-0.35355339…
Solve Using the Square Root Property (( square root of 5)/2)^2=((3( square root of 2))/4)^2+x^2