Precalculus Examples

$y^{\frac{9}{2}} - 10 y^{\frac{7}{2}} + 21 y^{\frac{5}{2}} = 0$

Step 1

Factor $y^{\frac{5}{2}}$ out of $y^{\frac{9}{2}} - 10 y^{\frac{7}{2}} + 21 y^{\frac{5}{2}}$ .

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Step 1.1

Factor $y^{\frac{5}{2}}$ out of $y^{\frac{9}{2}}$ .

$y^{\frac{5}{2}} y^{\frac{4}{2}} - 10 y^{\frac{7}{2}} + 21 y^{\frac{5}{2}} = 0$

Step 1.2

Factor $y^{\frac{5}{2}}$ out of $- 10 y^{\frac{7}{2}}$ .

$y^{\frac{5}{2}} y^{\frac{4}{2}} + y^{\frac{5}{2}} (- 10 y^{\frac{2}{2}}) + 21 y^{\frac{5}{2}} = 0$

Step 1.3

Factor $y^{\frac{5}{2}}$ out of $21 y^{\frac{5}{2}}$ .

$y^{\frac{5}{2}} y^{\frac{4}{2}} + y^{\frac{5}{2}} (- 10 y^{\frac{2}{2}}) + y^{\frac{5}{2}} \cdot 21 = 0$

Step 1.4

Factor $y^{\frac{5}{2}}$ out of $y^{\frac{5}{2}} y^{\frac{4}{2}} + y^{\frac{5}{2}} (- 10 y^{\frac{2}{2}})$ .

$y^{\frac{5}{2}} (y^{\frac{4}{2}} - 10 y^{\frac{2}{2}}) + y^{\frac{5}{2}} \cdot 21 = 0$

Step 1.5

Factor $y^{\frac{5}{2}}$ out of $y^{\frac{5}{2}} (y^{\frac{4}{2}} - 10 y^{\frac{2}{2}}) + y^{\frac{5}{2}} \cdot 21$ .

$y^{\frac{5}{2}} (y^{\frac{4}{2}} - 10 y^{\frac{2}{2}} + 21) = 0$

Step 2

Factor $y^{\frac{4}{2}} - 10 y^{\frac{2}{2}} + 21$ using the AC method.

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Step 2.1

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 $21$ and whose sum is $- 10$ .

$- 7, - 3$

Step 2.2

Write the factored form using these integers.

$y^{\frac{5}{2}} ((y^{\frac{2}{2}} - 7) (y^{\frac{2}{2}} - 3)) = 0$

Step 3

Divide $2$ by $2$ .

$y^{\frac{5}{2}} ((y - 7) (y^{\frac{2}{2}} - 3)) = 0$

Step 4

Divide $2$ by $2$ .

$y^{\frac{5}{2}} ((y - 7) (y - 3)) = 0$

Step 5

Remove unnecessary parentheses.

$y^{\frac{5}{2}} (y - 7) (y - 3) = 0$

Step 6

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

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

$y - 7 = 0$

$y - 3 = 0$

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

Simplify the exponent.

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Step 7.2.2.1

Simplify the left side.

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Step 7.2.2.1.1

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

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Step 7.2.2.1.1.1

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

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Step 7.2.2.1.1.1.1

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

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

Step 7.2.2.1.1.1.2

Cancel the common factor of $5$ .

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Step 7.2.2.1.1.1.2.1

Cancel the common factor.

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

Step 7.2.2.1.1.1.2.2

Rewrite the expression.

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

Step 7.2.2.1.1.1.3

Cancel the common factor of $2$ .

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Step 7.2.2.1.1.1.3.1

Cancel the common factor.

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

Step 7.2.2.1.1.1.3.2

Rewrite the expression.

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

Step 7.2.2.1.1.2

Simplify.

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

Step 7.2.2.2

Simplify the right side.

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Step 7.2.2.2.1

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

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Step 7.2.2.2.1.1

Simplify the expression.

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Step 7.2.2.2.1.1.1

Rewrite $0$ as $0^{5}$ .

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

Step 7.2.2.2.1.1.2

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

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

Step 7.2.2.2.1.2

Cancel the common factor of $5$ .

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Step 7.2.2.2.1.2.1

Cancel the common factor.

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

Step 7.2.2.2.1.2.2

Rewrite the expression.

$y = 0^{2}$

Step 7.2.2.2.1.3

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

$y = 0$

Step 8

Set $y - 7$ equal to $0$ and solve for $y$ .

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Step 8.1

Set $y - 7$ equal to $0$ .

$y - 7 = 0$

Step 8.2

Add $7$ to both sides of the equation.

$y = 7$

Step 9

Set $y - 3$ equal to $0$ and solve for $y$ .

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Step 9.1

Set $y - 3$ equal to $0$ .

$y - 3 = 0$

Step 9.2

Add $3$ to both sides of the equation.

$y = 3$

Step 10

The final solution is all the values that make $y^{\frac{5}{2}} (y - 7) (y - 3) = 0$ true.

$y = 0, 7, 3$