Solve for d 6d^(2/3)-11d^(1/3)-2=0

$6 d^{\frac{2}{3}} - 11 d^{\frac{1}{3}} - 2 = 0$

Step 1

Factor the left side of the equation.

Rewrite $d^{\frac{2}{3}}$ as ${(d^{\frac{1}{3}})}^{2}$ .

$6 {(d^{\frac{1}{3}})}^{2} - 11 d^{\frac{1}{3}} - 2 = 0$

Let $u = d^{\frac{1}{3}}$ . Substitute $u$ for all occurrences of $d^{\frac{1}{3}}$ .

$6 u^{2} - 11 u - 2 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 6 \cdot - 2 = - 12$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 u$ .

$6 u^{2} - 11 u - 2 = 0$

Rewrite $- 11$ as $1$ plus $- 12$

$6 u^{2} + (1 - 12) u - 2 = 0$

Apply the distributive property.

$6 u^{2} + 1 u - 12 u - 2 = 0$

Multiply $u$ by $1$ .

$6 u^{2} + u - 12 u - 2 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(6 u^{2} + u) - 12 u - 2 = 0$

Factor out the greatest common factor (GCF) from each group.

$u (6 u + 1) - 2 (6 u + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $6 u + 1$ .

$(6 u + 1) (u - 2) = 0$

Replace all occurrences of $u$ with $d^{\frac{1}{3}}$ .

$(6 d^{\frac{1}{3}} + 1) (d^{\frac{1}{3}} - 2) = 0$

Step 2

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

$6 d^{\frac{1}{3}} + 1 = 0$

$d^{\frac{1}{3}} - 2 = 0$

Step 3

Set $6 d^{\frac{1}{3}} + 1$ equal to $0$ and solve for $d$ .

Set $6 d^{\frac{1}{3}} + 1$ equal to $0$ .

$6 d^{\frac{1}{3}} + 1 = 0$

Solve $6 d^{\frac{1}{3}} + 1 = 0$ for $d$ .

Subtract $1$ from both sides of the equation.

$6 d^{\frac{1}{3}} = - 1$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(6 d^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${(6 d^{\frac{1}{3}})}^{3}$ .

Apply the product rule to $6 d^{\frac{1}{3}}$ .

$6^{3} {(d^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Raise $6$ to the power of $3$ .

$216 {(d^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Multiply the exponents in ${(d^{\frac{1}{3}})}^{3}$ .

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

$216 d^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$216 d^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Rewrite the expression.

$216 d^{1} = {(- 1)}^{3}$

Simplify.

$216 d = {(- 1)}^{3}$

Simplify the right side.

Raise $- 1$ to the power of $3$ .

$216 d = - 1$

Divide each term in $216 d = - 1$ by $216$ and simplify.

Divide each term in $216 d = - 1$ by $216$ .

$\frac{216 d}{216} = \frac{- 1}{216}$

Simplify the left side.

Cancel the common factor of $216$ .

Cancel the common factor.

$\frac{216 d}{216} = \frac{- 1}{216}$

Divide $d$ by $1$ .

$d = \frac{- 1}{216}$

Simplify the right side.

Move the negative in front of the fraction.

$d = - \frac{1}{216}$

Step 4

Set $d^{\frac{1}{3}} - 2$ equal to $0$ and solve for $d$ .

Set $d^{\frac{1}{3}} - 2$ equal to $0$ .

$d^{\frac{1}{3}} - 2 = 0$

Solve $d^{\frac{1}{3}} - 2 = 0$ for $d$ .

Add $2$ to both sides of the equation.

$d^{\frac{1}{3}} = 2$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(d^{\frac{1}{3}})}^{3} = 2^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${(d^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${(d^{\frac{1}{3}})}^{3}$ .

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

$d^{\frac{1}{3} \cdot 3} = 2^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$d^{\frac{1}{3} \cdot 3} = 2^{3}$

Rewrite the expression.

$d^{1} = 2^{3}$

Simplify.

$d = 2^{3}$

Simplify the right side.

Raise $2$ to the power of $3$ .

$d = 8$

Step 5

The final solution is all the values that make $(6 d^{\frac{1}{3}} + 1) (d^{\frac{1}{3}} - 2) = 0$ true.

$d = - \frac{1}{216}, 8$

Step 6

The result can be shown in multiple forms.

Exact Form:

$d = - \frac{1}{216}, 8$

Decimal Form:

$d = - 0.004\overline{629}, 8$

Solve for d 6d^(2/3)-11d^(1/3)-2=0

Related Questions

Support LCMGCF.com by disabling your adblocker.