Solve for n n^(1/2)+6n^(1/4)-16=0

$n^{\frac{1}{2}} + 6 n^{\frac{1}{4}} - 16 = 0$

Step 1

Factor the left side of the equation.

Rewrite $n^{\frac{1}{2}}$ as ${(n^{\frac{1}{4}})}^{2}$ .

${(n^{\frac{1}{4}})}^{2} + 6 n^{\frac{1}{4}} - 16 = 0$

Let $u = n^{\frac{1}{4}}$ . Substitute $u$ for all occurrences of $n^{\frac{1}{4}}$ .

$u^{2} + 6 u - 16 = 0$

Factor $u^{2} + 6 u - 16$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 16$ and whose sum is $6$ .

$- 2, 8$

Write the factored form using these integers.

$(u - 2) (u + 8) = 0$

Replace all occurrences of $u$ with $n^{\frac{1}{4}}$ .

$(n^{\frac{1}{4}} - 2) (n^{\frac{1}{4}} + 8) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$n^{\frac{1}{4}} - 2 = 0$

$n^{\frac{1}{4}} + 8 = 0$

Step 3

Set $n^{\frac{1}{4}} - 2$ equal to $0$ and solve for $n$ .

Set $n^{\frac{1}{4}} - 2$ equal to $0$ .

$n^{\frac{1}{4}} - 2 = 0$

Solve $n^{\frac{1}{4}} - 2 = 0$ for $n$ .

Add $2$ to both sides of the equation.

$n^{\frac{1}{4}} = 2$

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

${(n^{\frac{1}{4}})}^{4} = 2^{4}$

Simplify the exponent.

Simplify the left side.

Simplify ${(n^{\frac{1}{4}})}^{4}$ .

Multiply the exponents in ${(n^{\frac{1}{4}})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$n^{\frac{1}{4} \cdot 4} = 2^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$n^{\frac{1}{4} \cdot 4} = 2^{4}$

Rewrite the expression.

$n^{1} = 2^{4}$

Simplify.

$n = 2^{4}$

Simplify the right side.

Raise $2$ to the power of $4$ .

$n = 16$

Step 4

Set $n^{\frac{1}{4}} + 8$ equal to $0$ and solve for $n$ .

Set $n^{\frac{1}{4}} + 8$ equal to $0$ .

$n^{\frac{1}{4}} + 8 = 0$

Solve $n^{\frac{1}{4}} + 8 = 0$ for $n$ .

Subtract $8$ from both sides of the equation.

$n^{\frac{1}{4}} = - 8$

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

${(n^{\frac{1}{4}})}^{4} = {(- 8)}^{4}$

Simplify the exponent.

Simplify the left side.

Simplify ${(n^{\frac{1}{4}})}^{4}$ .

Multiply the exponents in ${(n^{\frac{1}{4}})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$n^{\frac{1}{4} \cdot 4} = {(- 8)}^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$n^{\frac{1}{4} \cdot 4} = {(- 8)}^{4}$

Rewrite the expression.

$n^{1} = {(- 8)}^{4}$

Simplify.

$n = {(- 8)}^{4}$

Simplify the right side.

Raise $- 8$ to the power of $4$ .

$n = 4096$

Step 5

The final solution is all the values that make $(n^{\frac{1}{4}} - 2) (n^{\frac{1}{4}} + 8) = 0$ true.

$n = 16, 4096$

Step 6

Exclude the solutions that do not make $n^{\frac{1}{2}} + 6 n^{\frac{1}{4}} - 16 = 0$ true.

$n = 16$

Solve for n n^(1/2)+6n^(1/4)-16=0

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