If the fractions are the exponents, then let's see what that means:
→ an exponent of 1/2 means you are taking the square root of the term
→ an exponent of -1/2 means you are taking the square root of the term and that the result is in the denominator of a fraction
→ an exponent of -3/2 is the same thing as -(1/2)(3), so you are taking the square root of the term, then cubing it, and that result will be in the denominator of a fraction
→ since there are no () around the number portions, only the variable x is affected by a negative exponent
Let's do some solving now!
First thing to do is get rid of the fractions in the exponents; to do that, square both sides
→ [x^(1/2) + 2x^(-1/2) = 63x^(-3/2)]^2
→ x^(1/2)(2) + 2x^(-1/2)(2) = 63x^(-3/2)(2)
→ x^(2/2) + 2x^(-2/2) = 63x^(-6/2)
→ x^1 + 2x^(-1) = 63x^(-3)
Now, clean that part up so we can see what to do next
→ x + 2/x = 63/x^3
Multiply everything by x^3 to get rid of the variables in the denominators
→ x^3[x + 2/x = 63/x^3]
→ (x^3)(x) + (x^3)(2/x) = (x^3)(63/x^3)
→ x^4 + (2x^3)/(x) = (63x^3)/(x^3)
→ x^4 + 2x^2 = 63
Collect all the terms on same side of the = sign
→ x^4 + 2x^2 - 63 = 0
Now, let's see if we can factor:
→ (x^2 + 9)(x^2 - 7)
Check using FOIL
F = first terms = (x^2)(x^2) = x^4
O = outer terms = (x^2)(-7) = -7x^2
I = inner terms = (9)(x^2) = 9x^2
L = last terms = (9)(-7) = -63
Collect all the terms
→ x^ - 7x^2 + 9x^2 - 63
→ x^4 + 2x^2 - 63 → they match!
Last, solve for each factor from above
→ x^2 + 9 = 0
→ x^2 = -9
→ x = ± square root (-9) → which are not necessarily real numbers
Answers & Comments
Verified answer
Use parentheses and "^" to indicate exponent.
x^(1/2) + 2x^(-1/2) = 63x^(-3/2)
x^(1/2) + 2x^(-1/2) - 63x^(-3/2) = 0
(x^(1/4) + 9x^(-3/4))(x^(1/4) - 7x^(-3/4)) = 0
x^(1/4) + 9x^(-3/4) = 0 or x^(1/4) - 7x^(-3/4) = 0
x^(1/4)(1 + 9x^-1) = 0 or x^(1/4)(1 - 7x^-1) = 0
x^(1/4) = 0 or 1 + 9/x = 0 or 1 - 7/x = 0
x = 0 or x = -9 or x = 7
Discard x = 0 because it leads to undefined numbers in the original equation.
Hello Nicole,
If the fractions are the exponents, then let's see what that means:
→ an exponent of 1/2 means you are taking the square root of the term
→ an exponent of -1/2 means you are taking the square root of the term and that the result is in the denominator of a fraction
→ an exponent of -3/2 is the same thing as -(1/2)(3), so you are taking the square root of the term, then cubing it, and that result will be in the denominator of a fraction
→ since there are no () around the number portions, only the variable x is affected by a negative exponent
Let's do some solving now!
First thing to do is get rid of the fractions in the exponents; to do that, square both sides
→ [x^(1/2) + 2x^(-1/2) = 63x^(-3/2)]^2
→ x^(1/2)(2) + 2x^(-1/2)(2) = 63x^(-3/2)(2)
→ x^(2/2) + 2x^(-2/2) = 63x^(-6/2)
→ x^1 + 2x^(-1) = 63x^(-3)
Now, clean that part up so we can see what to do next
→ x + 2/x = 63/x^3
Multiply everything by x^3 to get rid of the variables in the denominators
→ x^3[x + 2/x = 63/x^3]
→ (x^3)(x) + (x^3)(2/x) = (x^3)(63/x^3)
→ x^4 + (2x^3)/(x) = (63x^3)/(x^3)
→ x^4 + 2x^2 = 63
Collect all the terms on same side of the = sign
→ x^4 + 2x^2 - 63 = 0
Now, let's see if we can factor:
→ (x^2 + 9)(x^2 - 7)
Check using FOIL
F = first terms = (x^2)(x^2) = x^4
O = outer terms = (x^2)(-7) = -7x^2
I = inner terms = (9)(x^2) = 9x^2
L = last terms = (9)(-7) = -63
Collect all the terms
→ x^ - 7x^2 + 9x^2 - 63
→ x^4 + 2x^2 - 63 → they match!
Last, solve for each factor from above
→ x^2 + 9 = 0
→ x^2 = -9
→ x = ± square root (-9) → which are not necessarily real numbers
and
→ x^2 - 7 = 0
→ x^2 = 7
→ x = ± square root (7)
Thanks for an interesting question!
Jeffrey
x^1/2 + 2 / x^1/2 = 63 / x^3/2
Let x^1/2 be = a
a + 2/a = 63 / a^3
MULTIPLY BY a^3
a^4+ 2a^2= 63
a^4 + 2a^2 - 63 = 0
a^4 + 9a^2 -7a^2 -63 = 0
a^2 ( a^2 + 9) -7( a^2 +9) = 0
( a^2 - 7) ( a^2 +9) = 0
a^2 = 7 OR -9
x = 7 OR -9 ANSWER