Finite Math Examples

$2 y^{\frac{12}{5}} - 17 y^{\frac{7}{5}} + 35 y^{\frac{2}{5}} = 0$

Step 1

Find a common factor $y^{\frac{2}{5}}$ that is present in each term.

$2 {(y^{\frac{2}{5}})}^{6} - 17 {(y^{\frac{2}{5}})}^{\frac{7}{2}} + 35 y^{\frac{2}{5}}$

Step 2

Substitute $u$ for $y^{\frac{2}{5}}$ .

$2 {(u)}^{6} - 17 {(u)}^{\frac{7}{2}} + 35 (u) = 0$

Step 3

Solve for $u$ .

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Step 3.1

Factor the left side of the equation.

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Step 3.1.1

Factor $u$ out of $2 u^{6} - 17 u^{\frac{7}{2}} + 35 u$ .

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Step 3.1.1.1

Factor $u$ out of $2 u^{6}$ .

$u (2 u^{5}) - 17 u^{\frac{7}{2}} + 35 u = 0$

Step 3.1.1.2

Factor $u$ out of $- 17 u^{\frac{7}{2}}$ .

$u (2 u^{5}) + u (- 17 u^{\frac{5}{2}}) + 35 u = 0$

Step 3.1.1.3

Factor $u$ out of $35 u$ .

$u (2 u^{5}) + u (- 17 u^{\frac{5}{2}}) + u \cdot 35 = 0$

Step 3.1.1.4

Factor $u$ out of $u (2 u^{5}) + u (- 17 u^{\frac{5}{2}})$ .

$u (2 u^{5} - 17 u^{\frac{5}{2}}) + u \cdot 35 = 0$

Step 3.1.1.5

Factor $u$ out of $u (2 u^{5} - 17 u^{\frac{5}{2}}) + u \cdot 35$ .

$u (2 u^{5} - 17 u^{\frac{5}{2}} + 35) = 0$

Step 3.1.2

Rewrite $u^{5}$ as ${(u^{\frac{5}{2}})}^{2}$ .

$u (2 {(u^{\frac{5}{2}})}^{2} - 17 u^{\frac{5}{2}} + 35) = 0$

Step 3.1.3

Let $u = u^{\frac{5}{2}}$ . Substitute $u$ for all occurrences of $u^{\frac{5}{2}}$ .

$u (2 u^{2} - 17 u + 35) = 0$

Step 3.1.4

Factor by grouping.

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Step 3.1.4.1

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 $a \cdot c = 2 \cdot 35 = 70$ and whose sum is $b = - 17$ .

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Step 3.1.4.1.1

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

$u (2 u^{2} - 17 u + 35) = 0$

Step 3.1.4.1.2

Rewrite $- 17$ as $- 7$ plus $- 10$

$u (2 u^{2} + (- 7 - 10) u + 35) = 0$

Step 3.1.4.1.3

Apply the distributive property.

$u (2 u^{2} - 7 u - 10 u + 35) = 0$

Step 3.1.4.2

Factor out the greatest common factor from each group.

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Step 3.1.4.2.1

Group the first two terms and the last two terms.

$u ((2 u^{2} - 7 u) - 10 u + 35) = 0$

Step 3.1.4.2.2

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

$u (u (2 u - 7) - 5 (2 u - 7)) = 0$

Step 3.1.4.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 7$ .

$u ((2 u - 7) (u - 5)) = 0$

Step 3.1.5

Factor.

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Step 3.1.5.1

Replace all occurrences of $u$ with $u^{\frac{5}{2}}$ .

$u ((2 u^{\frac{5}{2}} - 7) (u^{\frac{5}{2}} - 5)) = 0$

Step 3.1.5.2

Remove unnecessary parentheses.

$u (2 u^{\frac{5}{2}} - 7) (u^{\frac{5}{2}} - 5) = 0$

Step 3.2

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

$u = 0$

$2 u^{\frac{5}{2}} - 7 = 0$

$u^{\frac{5}{2}} - 5 = 0$

Step 3.3

Set $u$ equal to $0$ .

$u = 0$

Step 3.4

Set $2 u^{\frac{5}{2}} - 7$ equal to $0$ and solve for $u$ .

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Step 3.4.1

Set $2 u^{\frac{5}{2}} - 7$ equal to $0$ .

$2 u^{\frac{5}{2}} - 7 = 0$

Step 3.4.2

Solve $2 u^{\frac{5}{2}} - 7 = 0$ for $u$ .

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Step 3.4.2.1

Add $7$ to both sides of the equation.

$2 u^{\frac{5}{2}} = 7$

Step 3.4.2.2

Raise each side of the equation to the power of $\frac{2}{5}$ to eliminate the fractional exponent on the left side.

${(2 u^{\frac{5}{2}})}^{\frac{2}{5}} = 7^{\frac{2}{5}}$

Step 3.4.2.3

Simplify the left side.

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Step 3.4.2.3.1

Simplify ${(2 u^{\frac{5}{2}})}^{\frac{2}{5}}$ .

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Step 3.4.2.3.1.1

Apply the product rule to $2 u^{\frac{5}{2}}$ .

$2^{\frac{2}{5}} {(u^{\frac{5}{2}})}^{\frac{2}{5}} = 7^{\frac{2}{5}}$

Step 3.4.2.3.1.2

Multiply the exponents in ${(u^{\frac{5}{2}})}^{\frac{2}{5}}$ .

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Step 3.4.2.3.1.2.1

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

$2^{\frac{2}{5}} u^{\frac{5}{2} \cdot \frac{2}{5}} = 7^{\frac{2}{5}}$

Step 3.4.2.3.1.2.2

Cancel the common factor of $5$ .

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Step 3.4.2.3.1.2.2.1

Cancel the common factor.

$2^{\frac{2}{5}} u^{\frac{5}{2} \cdot \frac{2}{5}} = 7^{\frac{2}{5}}$

Step 3.4.2.3.1.2.2.2

Rewrite the expression.

$2^{\frac{2}{5}} u^{\frac{1}{2} \cdot 2} = 7^{\frac{2}{5}}$

Step 3.4.2.3.1.2.3

Cancel the common factor of $2$ .

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Step 3.4.2.3.1.2.3.1

Cancel the common factor.

$2^{\frac{2}{5}} u^{\frac{1}{2} \cdot 2} = 7^{\frac{2}{5}}$

Step 3.4.2.3.1.2.3.2

Rewrite the expression.

$2^{\frac{2}{5}} u^{1} = 7^{\frac{2}{5}}$

Step 3.4.2.3.1.3

Simplify.

$2^{\frac{2}{5}} u = 7^{\frac{2}{5}}$

Step 3.4.2.3.1.4

Reorder factors in $2^{\frac{2}{5}} u$ .

$u \cdot 2^{\frac{2}{5}} = 7^{\frac{2}{5}}$

Step 3.4.2.4

Divide each term in $u \cdot 2^{\frac{2}{5}} = 7^{\frac{2}{5}}$ by $2^{\frac{2}{5}}$ and simplify.

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Step 3.4.2.4.1

Divide each term in $u \cdot 2^{\frac{2}{5}} = 7^{\frac{2}{5}}$ by $2^{\frac{2}{5}}$ .

$\frac{u \cdot 2^{\frac{2}{5}}}{2^{\frac{2}{5}}} = \frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}}$

Step 3.4.2.4.2

Simplify the left side.

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Step 3.4.2.4.2.1

Cancel the common factor.

$\frac{u \cdot 2^{\frac{2}{5}}}{2^{\frac{2}{5}}} = \frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}}$

Step 3.4.2.4.2.2

Divide $u$ by $1$ .

$u = \frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}}$

Step 3.5

Set $u^{\frac{5}{2}} - 5$ equal to $0$ and solve for $u$ .

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Step 3.5.1

Set $u^{\frac{5}{2}} - 5$ equal to $0$ .

$u^{\frac{5}{2}} - 5 = 0$

Step 3.5.2

Solve $u^{\frac{5}{2}} - 5 = 0$ for $u$ .

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Step 3.5.2.1

Add $5$ to both sides of the equation.

$u^{\frac{5}{2}} = 5$

Step 3.5.2.2

Raise each side of the equation to the power of $\frac{2}{5}$ to eliminate the fractional exponent on the left side.

${(u^{\frac{5}{2}})}^{\frac{2}{5}} = 5^{\frac{2}{5}}$

Step 3.5.2.3

Simplify the left side.

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Step 3.5.2.3.1

Simplify ${(u^{\frac{5}{2}})}^{\frac{2}{5}}$ .

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Step 3.5.2.3.1.1

Multiply the exponents in ${(u^{\frac{5}{2}})}^{\frac{2}{5}}$ .

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Step 3.5.2.3.1.1.1

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

$u^{\frac{5}{2} \cdot \frac{2}{5}} = 5^{\frac{2}{5}}$

Step 3.5.2.3.1.1.2

Cancel the common factor of $5$ .

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Step 3.5.2.3.1.1.2.1

Cancel the common factor.

$u^{\frac{5}{2} \cdot \frac{2}{5}} = 5^{\frac{2}{5}}$

Step 3.5.2.3.1.1.2.2

Rewrite the expression.

$u^{\frac{1}{2} \cdot 2} = 5^{\frac{2}{5}}$

Step 3.5.2.3.1.1.3

Cancel the common factor of $2$ .

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Step 3.5.2.3.1.1.3.1

Cancel the common factor.

$u^{\frac{1}{2} \cdot 2} = 5^{\frac{2}{5}}$

Step 3.5.2.3.1.1.3.2

Rewrite the expression.

$u^{1} = 5^{\frac{2}{5}}$

Step 3.5.2.3.1.2

Simplify.

$u = 5^{\frac{2}{5}}$

Step 3.6

The final solution is all the values that make $u (2 u^{\frac{5}{2}} - 7) (u^{\frac{5}{2}} - 5) = 0$ true.

$u = 0, \frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}}, 5^{\frac{2}{5}}$

Step 4

Substitute $y$ for $u$ .

$y^{\frac{2}{5}} = 0, \frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}}, 5^{\frac{2}{5}}$

Step 5

Solve for $y^{\frac{2}{5}} = 0$ for $y$ .

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Step 5.1

Raise each side of the equation to the power of $\frac{5}{2}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{2}{5}})}^{\frac{5}{2}} = \pm 0^{\frac{5}{2}}$

Step 5.2

Simplify the exponent.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify ${(y^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 5.2.1.1.1

Multiply the exponents in ${(y^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 5.2.1.1.1.1

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

$y^{\frac{2}{5} \cdot \frac{5}{2}} = \pm 0^{\frac{5}{2}}$

Step 5.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.1.2.1

Cancel the common factor.

$y^{\frac{2}{5} \cdot \frac{5}{2}} = \pm 0^{\frac{5}{2}}$

Step 5.2.1.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{5} \cdot 5} = \pm 0^{\frac{5}{2}}$

Step 5.2.1.1.1.3

Cancel the common factor of $5$ .

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Step 5.2.1.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{5} \cdot 5} = \pm 0^{\frac{5}{2}}$

Step 5.2.1.1.1.3.2

Rewrite the expression.

$y^{1} = \pm 0^{\frac{5}{2}}$

Step 5.2.1.1.2

Simplify.

$y = \pm 0^{\frac{5}{2}}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\pm 0^{\frac{5}{2}}$ .

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Step 5.2.2.1.1

Simplify the expression.

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Step 5.2.2.1.1.1

Rewrite $0$ as $0^{2}$ .

$y = \pm {(0^{2})}^{\frac{5}{2}}$

Step 5.2.2.1.1.2

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

$y = \pm 0^{2 (\frac{5}{2})}$

Step 5.2.2.1.2

Cancel the common factor of $2$ .

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Step 5.2.2.1.2.1

Cancel the common factor.

$y = \pm 0^{2 (\frac{5}{2})}$

Step 5.2.2.1.2.2

Rewrite the expression.

$y = \pm 0^{5}$

Step 5.2.2.1.3

Raising $0$ to any positive power yields $0$ .

$y = \pm 0$

Step 5.2.2.1.4

Plus or minus $0$ is $0$ .

$y = 0$

Step 6

Solve for $y^{\frac{2}{5}} = \frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}}$ for $y$ .

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Step 6.1

Raise each side of the equation to the power of $\frac{5}{2}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{2}{5}})}^{\frac{5}{2}} = \pm {(\frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}})}^{\frac{5}{2}}$

Step 6.2

Simplify the exponent.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify ${(y^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 6.2.1.1.1

Multiply the exponents in ${(y^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 6.2.1.1.1.1

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

$y^{\frac{2}{5} \cdot \frac{5}{2}} = \pm {(\frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}})}^{\frac{5}{2}}$

Step 6.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.1.1.1.2.1

Cancel the common factor.

$y^{\frac{2}{5} \cdot \frac{5}{2}} = \pm {(\frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}})}^{\frac{5}{2}}$

Step 6.2.1.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{5} \cdot 5} = \pm {(\frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}})}^{\frac{5}{2}}$

Step 6.2.1.1.1.3

Cancel the common factor of $5$ .

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Step 6.2.1.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{5} \cdot 5} = \pm {(\frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}})}^{\frac{5}{2}}$

Step 6.2.1.1.1.3.2

Rewrite the expression.

$y^{1} = \pm {(\frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}})}^{\frac{5}{2}}$

Step 6.2.1.1.2

Simplify.

$y = \pm {(\frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}})}^{\frac{5}{2}}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $\pm {(\frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}})}^{\frac{5}{2}}$ .

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Step 6.2.2.1.1

Apply the product rule to $\frac{7^{\frac{2}{5}}}{2^{\frac{2}{5}}}$ .

$y = \pm \frac{{(7^{\frac{2}{5}})}^{\frac{5}{2}}}{{(2^{\frac{2}{5}})}^{\frac{5}{2}}}$

Step 6.2.2.1.2

Simplify the numerator.

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Step 6.2.2.1.2.1

Multiply the exponents in ${(7^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 6.2.2.1.2.1.1

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

$y = \pm \frac{7^{\frac{2}{5} \cdot \frac{5}{2}}}{{(2^{\frac{2}{5}})}^{\frac{5}{2}}}$

Step 6.2.2.1.2.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.2.1.2.1

Cancel the common factor.

$y = \pm \frac{7^{\frac{2}{5} \cdot \frac{5}{2}}}{{(2^{\frac{2}{5}})}^{\frac{5}{2}}}$

Step 6.2.2.1.2.1.2.2

Rewrite the expression.

$y = \pm \frac{7^{\frac{1}{5} \cdot 5}}{{(2^{\frac{2}{5}})}^{\frac{5}{2}}}$

Step 6.2.2.1.2.1.3

Cancel the common factor of $5$ .

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Step 6.2.2.1.2.1.3.1

Cancel the common factor.

$y = \pm \frac{7^{\frac{1}{5} \cdot 5}}{{(2^{\frac{2}{5}})}^{\frac{5}{2}}}$

Step 6.2.2.1.2.1.3.2

Rewrite the expression.

$y = \pm \frac{7^{1}}{{(2^{\frac{2}{5}})}^{\frac{5}{2}}}$

Step 6.2.2.1.2.2

Evaluate the exponent.

$y = \pm \frac{7}{{(2^{\frac{2}{5}})}^{\frac{5}{2}}}$

Step 6.2.2.1.3

Simplify the denominator.

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Step 6.2.2.1.3.1

Multiply the exponents in ${(2^{\frac{2}{5}})}^{\frac{5}{2}}$ .

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Step 6.2.2.1.3.1.1

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

$y = \pm \frac{7}{2^{\frac{2}{5} \cdot \frac{5}{2}}}$

Step 6.2.2.1.3.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.3.1.2.1

Cancel the common factor.

$y = \pm \frac{7}{2^{\frac{2}{5} \cdot \frac{5}{2}}}$

Step 6.2.2.1.3.1.2.2

Rewrite the expression.

$y = \pm \frac{7}{2^{\frac{1}{5} \cdot 5}}$

Step 6.2.2.1.3.1.3

Cancel the common factor of $5$ .

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Step 6.2.2.1.3.1.3.1

Cancel the common factor.

$y = \pm \frac{7}{2^{\frac{1}{5} \cdot 5}}$

Step 6.2.2.1.3.1.3.2

Rewrite the expression.

$y = \pm \frac{7}{2^{1}}$

Step 6.2.2.1.3.2

Evaluate the exponent.

$y = \pm \frac{7}{2}$

Step 6.3

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

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Step 6.3.1

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

$y = \frac{7}{2}$

Step 6.3.2

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

$y = - \frac{7}{2}$

Step 6.3.3

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

$y = \frac{7}{2}, - \frac{7}{2}$

Step 7

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$y = 5$

Step 8

List all of the solutions.

$y = 0, \frac{7}{2}, - \frac{7}{2}, 5$

Step 9

Exclude the solutions that do not make $2 y^{\frac{12}{5}} - 17 y^{\frac{7}{5}} + 35 y^{\frac{2}{5}} = 0$ true.

$y = 0, \frac{7}{2}, 5$

Step 10

The result can be shown in multiple forms.

Exact Form:

$y = 0, \frac{7}{2}, 5$

Decimal Form:

$y = 0, 3.5, 5$

Mixed Number Form:

$y = 0, 3 \frac{1}{2}, 5$