Basic Math Examples

$4 k^{\frac{4}{3}} - 65 k^{\frac{2}{3}} + 16 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

$4 {(k^{\frac{2}{3}})}^{2} - 65 k^{\frac{2}{3}} + 16 = 0$

Step 1.2

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

$4 u^{2} - 65 u + 16 = 0$

Step 1.3

Factor by grouping.

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Step 1.3.1

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 $a \cdot c = 4 \cdot 16 = 64$ and whose sum is $b = - 65$ .

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Step 1.3.1.1

Factor $- 65$ out of $- 65 u$ .

$4 u^{2} - 65 u + 16 = 0$

Step 1.3.1.2

Rewrite $- 65$ as $- 1$ plus $- 64$

$4 u^{2} + (- 1 - 64) u + 16 = 0$

Step 1.3.1.3

Apply the distributive property.

$4 u^{2} - 1 u - 64 u + 16 = 0$

Step 1.3.2

Factor out the greatest common factor from each group.

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Step 1.3.2.1

Group the first two terms and the last two terms.

$(4 u^{2} - 1 u) - 64 u + 16 = 0$

Step 1.3.2.2

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

$u (4 u - 1) - 16 (4 u - 1) = 0$

Step 1.3.3

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

$(4 u - 1) (u - 16) = 0$

Step 1.4

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

$(4 k^{\frac{2}{3}} - 1) (k^{\frac{2}{3}} - 16) = 0$

Step 2

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

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

$k^{\frac{2}{3}} - 16 = 0$

Step 3

Set $4 k^{\frac{2}{3}} - 1$ equal to $0$ and solve for $k$ .

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Step 3.1

Set $4 k^{\frac{2}{3}} - 1$ equal to $0$ .

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

Step 3.2

Solve $4 k^{\frac{2}{3}} - 1 = 0$ for $k$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

$4 k^{\frac{2}{3}} = 1$

Step 3.2.2

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

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

Step 3.2.3

Simplify the exponent.

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Step 3.2.3.1

Simplify the left side.

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Step 3.2.3.1.1

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

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Step 3.2.3.1.1.1

Simplify the expression.

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Step 3.2.3.1.1.1.1

Apply the product rule to $4 k^{\frac{2}{3}}$ .

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

Step 3.2.3.1.1.1.2

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

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

Step 3.2.3.1.1.1.3

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

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

Step 3.2.3.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.3.1.1.2.1

Cancel the common factor.

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

Step 3.2.3.1.1.2.2

Rewrite the expression.

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

Step 3.2.3.1.1.3

Simplify the expression.

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Step 3.2.3.1.1.3.1

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

$8 {(k^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 1^{\frac{3}{2}}$

Step 3.2.3.1.1.3.2

Multiply the exponents in ${(k^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 3.2.3.1.1.3.2.1

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

$8 k^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 1^{\frac{3}{2}}$

Step 3.2.3.1.1.3.2.2

Cancel the common factor of $2$ .

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Step 3.2.3.1.1.3.2.2.1

Cancel the common factor.

$8 k^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 1^{\frac{3}{2}}$

Step 3.2.3.1.1.3.2.2.2

Rewrite the expression.

$8 k^{\frac{1}{3} \cdot 3} = \pm 1^{\frac{3}{2}}$

Step 3.2.3.1.1.3.2.3

Cancel the common factor of $3$ .

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Step 3.2.3.1.1.3.2.3.1

Cancel the common factor.

$8 k^{\frac{1}{3} \cdot 3} = \pm 1^{\frac{3}{2}}$

Step 3.2.3.1.1.3.2.3.2

Rewrite the expression.

$8 k^{1} = \pm 1^{\frac{3}{2}}$

Step 3.2.3.1.1.4

Simplify.

$8 k = \pm 1^{\frac{3}{2}}$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

One to any power is one.

$8 k = \pm 1$

Step 3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$8 k = 1$

Step 3.2.4.2

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

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Step 3.2.4.2.1

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

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

Step 3.2.4.2.2

Simplify the left side.

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Step 3.2.4.2.2.1

Cancel the common factor of $8$ .

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Step 3.2.4.2.2.1.1

Cancel the common factor.

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

Step 3.2.4.2.2.1.2

Divide $k$ by $1$ .

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

Step 3.2.4.3

Next, use the negative value of the $\pm$ to find the second solution.

$8 k = - 1$

Step 3.2.4.4

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

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Step 3.2.4.4.1

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

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

Step 3.2.4.4.2

Simplify the left side.

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Step 3.2.4.4.2.1

Cancel the common factor of $8$ .

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Step 3.2.4.4.2.1.1

Cancel the common factor.

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

Step 3.2.4.4.2.1.2

Divide $k$ by $1$ .

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

Step 3.2.4.4.3

Simplify the right side.

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Step 3.2.4.4.3.1

Move the negative in front of the fraction.

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

Step 3.2.4.5

The complete solution is the result of both the positive and negative portions of the solution.

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

Step 4

Set $k^{\frac{2}{3}} - 16$ equal to $0$ and solve for $k$ .

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Step 4.1

Set $k^{\frac{2}{3}} - 16$ equal to $0$ .

$k^{\frac{2}{3}} - 16 = 0$

Step 4.2

Solve $k^{\frac{2}{3}} - 16 = 0$ for $k$ .

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Step 4.2.1

Add $16$ to both sides of the equation.

$k^{\frac{2}{3}} = 16$

Step 4.2.2

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(k^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 16^{\frac{3}{2}}$

Step 4.2.3

Simplify the exponent.

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Step 4.2.3.1

Simplify the left side.

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Step 4.2.3.1.1

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

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Step 4.2.3.1.1.1

Multiply the exponents in ${(k^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 4.2.3.1.1.1.1

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

$k^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 16^{\frac{3}{2}}$

Step 4.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.3.1.1.1.2.1

Cancel the common factor.

$k^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 16^{\frac{3}{2}}$

Step 4.2.3.1.1.1.2.2

Rewrite the expression.

$k^{\frac{1}{3} \cdot 3} = \pm 16^{\frac{3}{2}}$

Step 4.2.3.1.1.1.3

Cancel the common factor of $3$ .

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Step 4.2.3.1.1.1.3.1

Cancel the common factor.

$k^{\frac{1}{3} \cdot 3} = \pm 16^{\frac{3}{2}}$

Step 4.2.3.1.1.1.3.2

Rewrite the expression.

$k^{1} = \pm 16^{\frac{3}{2}}$

Step 4.2.3.1.1.2

Simplify.

$k = \pm 16^{\frac{3}{2}}$

Step 4.2.3.2

Simplify the right side.

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Step 4.2.3.2.1

Simplify $\pm 16^{\frac{3}{2}}$ .

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Step 4.2.3.2.1.1

Simplify the expression.

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Step 4.2.3.2.1.1.1

Rewrite $16$ as $4^{2}$ .

$k = \pm {(4^{2})}^{\frac{3}{2}}$

Step 4.2.3.2.1.1.2

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

$k = \pm 4^{2 (\frac{3}{2})}$

Step 4.2.3.2.1.2

Cancel the common factor of $2$ .

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Step 4.2.3.2.1.2.1

Cancel the common factor.

$k = \pm 4^{2 (\frac{3}{2})}$

Step 4.2.3.2.1.2.2

Rewrite the expression.

$k = \pm 4^{3}$

Step 4.2.3.2.1.3

Raise $4$ to the power of $3$ .

$k = \pm 64$

Step 4.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$k = 64$

Step 4.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$k = - 64$

Step 4.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$k = 64, - 64$

Step 5

The final solution is all the values that make $(4 k^{\frac{2}{3}} - 1) (k^{\frac{2}{3}} - 16) = 0$ true.

$k = \frac{1}{8}, - \frac{1}{8}, 64, - 64$

Step 6

The result can be shown in multiple forms.

Exact Form:

$k = \frac{1}{8}, - \frac{1}{8}, 64, - 64$

Decimal Form:

$k = 0.125, - 0.125, 64, - 64$