Algebra Examples

$4 k^{\frac{4}{3}} - 17 k^{\frac{2}{3}} + 4 = 0$

Step 1

Rewrite $k^{\frac{4}{3}}$ as ${(k^{\frac{2}{3}})}^{2}$ .

$4 {(k^{\frac{2}{3}})}^{2} - 17 k^{\frac{2}{3}} + 4 = 0$

Step 2

Let $u = k^{\frac{2}{3}}$ . Substitute $u$ for all occurrences of $k^{\frac{2}{3}}$ .

$4 u^{2} - 17 u + 4 = 0$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 4 \cdot 4 = 16$ and whose sum is $b = - 17$ .

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Step 3.1.1

Factor $- 17$ out of $- 17 u$ .

$4 u^{2} - 17 u + 4 = 0$

Step 3.1.2

Rewrite $- 17$ as $- 1$ plus $- 16$

$4 u^{2} + (- 1 - 16) u + 4 = 0$

Step 3.1.3

Apply the distributive property.

$4 u^{2} - 1 u - 16 u + 4 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(4 u^{2} - 1 u) - 16 u + 4 = 0$

Step 3.2.2

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

$u (4 u - 1) - 4 (4 u - 1) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $4 u - 1$ .

$(4 u - 1) (u - 4) = 0$

Step 4

Replace all occurrences of $u$ with $k^{\frac{2}{3}}$ .

$(4 k^{\frac{2}{3}} - 1) (k^{\frac{2}{3}} - 4) = 0$

Step 5

Rewrite $4 k^{\frac{2}{3}}$ as ${(2 k^{\frac{1}{3}})}^{2}$ .

$({(2 k^{\frac{1}{3}})}^{2} - 1) (k^{\frac{2}{3}} - 4) = 0$

Step 6

Rewrite $1$ as $1^{2}$ .

$({(2 k^{\frac{1}{3}})}^{2} - 1^{2}) (k^{\frac{2}{3}} - 4) = 0$

Step 7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 k^{\frac{1}{3}}$ and $b = 1$ .

$(2 k^{\frac{1}{3}} + 1) (2 k^{\frac{1}{3}} - 1) (k^{\frac{2}{3}} - 4) = 0$

Step 8

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

$(2 k^{\frac{1}{3}} + 1) (2 k^{\frac{1}{3}} - 1) ({(k^{\frac{1}{3}})}^{2} - 4) = 0$

Step 9

Rewrite $4$ as $2^{2}$ .

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

Step 10

Factor.

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Step 10.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = k^{\frac{1}{3}}$ and $b = 2$ .

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

Step 10.2

Remove unnecessary parentheses.

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

Step 11

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

$2 k^{\frac{1}{3}} + 1 = 0$

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

$k^{\frac{1}{3}} + 2 = 0$

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

Step 12

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

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Step 12.1

Set $2 k^{\frac{1}{3}} + 1$ equal to $0$ .

$2 k^{\frac{1}{3}} + 1 = 0$

Step 12.2

Solve $2 k^{\frac{1}{3}} + 1 = 0$ for $k$ .

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Step 12.2.1

Subtract $1$ from both sides of the equation.

$2 k^{\frac{1}{3}} = - 1$

Step 12.2.2

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

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

Step 12.2.3

Simplify the exponent.

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Step 12.2.3.1

Simplify the left side.

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Step 12.2.3.1.1

Simplify ${(2 k^{\frac{1}{3}})}^{3}$ .

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Step 12.2.3.1.1.1

Apply the product rule to $2 k^{\frac{1}{3}}$ .

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

Step 12.2.3.1.1.2

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

$8 {(k^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 12.2.3.1.1.3

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

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Step 12.2.3.1.1.3.1

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

$8 k^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 12.2.3.1.1.3.2

Cancel the common factor of $3$ .

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Step 12.2.3.1.1.3.2.1

Cancel the common factor.

$8 k^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 12.2.3.1.1.3.2.2

Rewrite the expression.

$8 k^{1} = {(- 1)}^{3}$

Step 12.2.3.1.1.4

Simplify.

$8 k = {(- 1)}^{3}$

Step 12.2.3.2

Simplify the right side.

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Step 12.2.3.2.1

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

$8 k = - 1$

Step 12.2.4

Divide each term in $8 k = - 1$ by $8$ and simplify.

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Step 12.2.4.1

Divide each term in $8 k = - 1$ by $8$ .

$\frac{8 k}{8} = \frac{- 1}{8}$

Step 12.2.4.2

Simplify the left side.

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Step 12.2.4.2.1

Cancel the common factor of $8$ .

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Step 12.2.4.2.1.1

Cancel the common factor.

$\frac{8 k}{8} = \frac{- 1}{8}$

Step 12.2.4.2.1.2

Divide $k$ by $1$ .

$k = \frac{- 1}{8}$

Step 12.2.4.3

Simplify the right side.

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Step 12.2.4.3.1

Move the negative in front of the fraction.

$k = - \frac{1}{8}$

Step 13

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

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Step 13.1

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

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

Step 13.2

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

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Step 13.2.1

Add $1$ to both sides of the equation.

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

Step 13.2.2

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

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

Step 13.2.3

Simplify the exponent.

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Step 13.2.3.1

Simplify the left side.

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Step 13.2.3.1.1

Simplify ${(2 k^{\frac{1}{3}})}^{3}$ .

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Step 13.2.3.1.1.1

Apply the product rule to $2 k^{\frac{1}{3}}$ .

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

Step 13.2.3.1.1.2

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

$8 {(k^{\frac{1}{3}})}^{3} = 1^{3}$

Step 13.2.3.1.1.3

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

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Step 13.2.3.1.1.3.1

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

$8 k^{\frac{1}{3} \cdot 3} = 1^{3}$

Step 13.2.3.1.1.3.2

Cancel the common factor of $3$ .

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Step 13.2.3.1.1.3.2.1

Cancel the common factor.

$8 k^{\frac{1}{3} \cdot 3} = 1^{3}$

Step 13.2.3.1.1.3.2.2

Rewrite the expression.

$8 k^{1} = 1^{3}$

Step 13.2.3.1.1.4

Simplify.

$8 k = 1^{3}$

Step 13.2.3.2

Simplify the right side.

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Step 13.2.3.2.1

One to any power is one.

$8 k = 1$

Step 13.2.4

Divide each term in $8 k = 1$ by $8$ and simplify.

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Step 13.2.4.1

Divide each term in $8 k = 1$ by $8$ .

$\frac{8 k}{8} = \frac{1}{8}$

Step 13.2.4.2

Simplify the left side.

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Step 13.2.4.2.1

Cancel the common factor of $8$ .

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Step 13.2.4.2.1.1

Cancel the common factor.

$\frac{8 k}{8} = \frac{1}{8}$

Step 13.2.4.2.1.2

Divide $k$ by $1$ .

$k = \frac{1}{8}$

Step 14

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

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Step 14.1

Set $k^{\frac{1}{3}} + 2$ equal to $0$ .

$k^{\frac{1}{3}} + 2 = 0$

Step 14.2

Solve $k^{\frac{1}{3}} + 2 = 0$ for $k$ .

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Step 14.2.1

Subtract $2$ from both sides of the equation.

$k^{\frac{1}{3}} = - 2$

Step 14.2.2

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

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

Step 14.2.3

Simplify the exponent.

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Step 14.2.3.1

Simplify the left side.

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Step 14.2.3.1.1

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

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Step 14.2.3.1.1.1

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

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Step 14.2.3.1.1.1.1

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

$k^{\frac{1}{3} \cdot 3} = {(- 2)}^{3}$

Step 14.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 14.2.3.1.1.1.2.1

Cancel the common factor.

$k^{\frac{1}{3} \cdot 3} = {(- 2)}^{3}$

Step 14.2.3.1.1.1.2.2

Rewrite the expression.

$k^{1} = {(- 2)}^{3}$

Step 14.2.3.1.1.2

Simplify.

$k = {(- 2)}^{3}$

Step 14.2.3.2

Simplify the right side.

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Step 14.2.3.2.1

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

$k = - 8$

Step 15

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

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Step 15.1

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

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

Step 15.2

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

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Step 15.2.1

Add $2$ to both sides of the equation.

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

Step 15.2.2

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

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

Step 15.2.3

Simplify the exponent.

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Step 15.2.3.1

Simplify the left side.

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Step 15.2.3.1.1

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

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Step 15.2.3.1.1.1

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

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Step 15.2.3.1.1.1.1

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

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

Step 15.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 15.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 15.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 15.2.3.1.1.2

Simplify.

$k = 2^{3}$

Step 15.2.3.2

Simplify the right side.

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Step 15.2.3.2.1

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

$k = 8$

Step 16

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

$k = - \frac{1}{8}, \frac{1}{8}, - 8, 8$