Solve (5y)/(y^(2) 4y^4) / (20y^(2))/(y^(2)-4)

Question

Simplify the expression

\frac{y ^{2} - 4}{16 y ^{7}}

Evaluate

\frac{\frac{5 y}{y ^{2} \times 4 y ^{4}}}{\frac{20 y ^{2}}{y ^{2} - 4}}

Reduce the fraction

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Evaluate

\frac{5 y}{y ^{2} \times 4 y ^{4}}

Multiply

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Evaluate

y^{2} \times 4 y^{4}

Multiply the terms with the same base by adding their exponents

y^{2 + 4} \times 4

Add the numbers

y^{6} \times 4

\frac{5 y}{y ^{6} \times 4}

Reduce the fraction

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Calculate

\frac{y}{y ^{6}}

Use the product rule \frac{a ^{n}}{a ^{m}} = a^{n - m} to simplify the expression

\frac{1}{y ^{6 - 1}}

Subtract the terms

\frac{1}{y ^{5}}

\frac{5}{y ^{5} \times 4}

Calculate

\frac{5}{4 y ^{5}}

\frac{\frac{5}{4 y ^{5}}}{\frac{20 y ^{2}}{y ^{2} - 4}}

Multiply by the reciprocal

\frac{5}{4 y ^{5}} \times \frac{y ^{2} - 4}{20 y ^{2}}

Cancel out the common factor 5

\frac{1}{4 y ^{5}} \times \frac{y ^{2} - 4}{4 y ^{2}}

Multiply the terms

\frac{y ^{2} - 4}{4 y ^{5} \times 4 y ^{2}}

Solution

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Evaluate

4 y^{5} \times 4 y^{2}

Multiply the numbers

16 y^{5} \times y^{2}

Multiply the terms

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Evaluate

y^{5} \times y^{2}

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

y^{5 + 2}

Add the numbers

y^{7}

16 y^{7}

\frac{y ^{2} - 4}{16 y ^{7}}

Show Solution

Find the excluded values

y = 0, y = 2, y = - 2

Evaluate

\frac{\frac{5 y}{y ^{2} \times 4 y ^{4}}}{\frac{20 y ^{2}}{y ^{2} - 4}}

To find the excluded values,set the denominators equal to 0

y^{2} \times y^{4} = 0 y^{2} - 4 = 0 \frac{20 y ^{2}}{y ^{2} - 4} = 0

Solve the equations

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Evaluate

y^{2} \times y^{4} = 0

Multiply the terms

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Evaluate

y^{2} \times y^{4}

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

y^{2 + 4}

Add the numbers

y^{6}

y^{6} = 0

The only way a power can be 0 is when the base equals 0

y = 0

y = 0 y^{2} - 4 = 0 \frac{20 y ^{2}}{y ^{2} - 4} = 0

Solve the equations

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Evaluate

y^{2} - 4 = 0

Move the constant to the right-hand side and change its sign

y^{2} = 0 + 4

Removing 0 doesn't change the value,so remove it from the expression

y^{2} = 4

Take the root of both sides of the equation and remember to use both positive and negative roots

y = \pm 4

Simplify the expression

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Evaluate

4

Write the number in exponential form with the base of 2

2^{2}

Reduce the index of the radical and exponent with 2

2

y = \pm 2

Separate the equation into 2 possible cases

y = 2 y = - 2

y = 0 y = 2 y = - 2 \frac{20 y ^{2}}{y ^{2} - 4} = 0

Solve the equations

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Evaluate

\frac{20 y ^{2}}{y ^{2} - 4} = 0

Cross multiply

20 y^{2} = (y^{2} - 4) \times 0

Simplify the equation

20 y^{2} = 0

Rewrite the expression

y^{2} = 0

The only way a power can be 0 is when the base equals 0

y = 0

y = 0 y = 2 y = - 2 y = 0

Solution

y = 0, y = 2, y = - 2

Show Solution

Find the roots

y \in \emptyset

Evaluate

\frac{\frac{5 y}{y ^{2} \times 4 y ^{4}}}{\frac{20 y ^{2}}{y ^{2} - 4}}

To find the roots of the expression,set the expression equal to 0

\frac{\frac{5 y}{y ^{2} \times 4 y ^{4}}}{\frac{20 y ^{2}}{y ^{2} - 4}} = 0

Find the domain

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Evaluate

⎩ ⎨ ⎧ y^{2} \times y^{4} \neq = 0 y^{2} - 4 \neq = 0 \frac{20 y ^{2}}{y ^{2} - 4} \neq = 0

Calculate

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Evaluate

y^{2} \times y^{4} \neq = 0

Multiply the terms

y^{6} \neq = 0

The only way a power can not be 0 is when the base not equals 0

y \neq = 0

⎩ ⎨ ⎧ y \neq = 0 y^{2} - 4 \neq = 0 \frac{20 y ^{2}}{y ^{2} - 4} \neq = 0

Calculate

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Evaluate

y^{2} - 4 \neq = 0

Move the constant to the right side

y^{2} \neq = 4

Take the root of both sides of the equation and remember to use both positive and negative roots

y \neq = \pm 4

Simplify the expression

y \neq = \pm 2

Separate the inequality into 2 possible cases

{y \neq = 2 y \neq = - 2

Find the intersection

y \in (- \infty, - 2) \cup (- 2, 2) \cup (2, + \infty)

⎩ ⎨ ⎧ y \neq = 0 y \in (- \infty, - 2) \cup (- 2, 2) \cup (2, + \infty) \frac{20 y ^{2}}{y ^{2} - 4} \neq = 0

Calculate

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Evaluate

\frac{20 y ^{2}}{y ^{2} - 4} \neq = 0

Multiply both sides

\frac{20 y ^{2}}{y ^{2} - 4} \times (y^{2} - 4) \neq = 0 \times (y^{2} - 4)

Evaluate

20 y^{2} \neq = 0 \times (y^{2} - 4)

Multiply both sides

20 y^{2} \neq = 0

Rewrite the expression

y^{2} \neq = 0

The only way a power can not be 0 is when the base not equals 0

y \neq = 0

⎩ ⎨ ⎧ y \neq = 0 y \in (- \infty, - 2) \cup (- 2, 2) \cup (2, + \infty) y \neq = 0

Simplify

{y \neq = 0 y \in (- \infty, - 2) \cup (- 2, 2) \cup (2, + \infty)

Find the intersection

y \in (- \infty, - 2) \cup (- 2, 0) \cup (0, 2) \cup (2, + \infty)

\frac{\frac{5 y}{y ^{2} \times 4 y ^{4}}}{\frac{20 y ^{2}}{y ^{2} - 4}} = 0, y \in (- \infty, - 2) \cup (- 2, 0) \cup (0, 2) \cup (2, + \infty)

Calculate

\frac{\frac{5 y}{y ^{2} \times 4 y ^{4}}}{\frac{20 y ^{2}}{y ^{2} - 4}} = 0

Multiply

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Multiply the terms

y^{2} \times 4 y^{4}

Multiply the terms with the same base by adding their exponents

y^{2 + 4} \times 4

Add the numbers

y^{6} \times 4

Use the commutative property to reorder the terms

4 y^{6}

\frac{\frac{5 y}{4 y ^{6}}}{\frac{20 y ^{2}}{y ^{2} - 4}} = 0

Divide the terms

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Evaluate

\frac{5 y}{4 y ^{6}}

Use the product rule \frac{a ^{n}}{a ^{m}} = a^{n - m} to simplify the expression

\frac{5}{4 y ^{6 - 1}}

Reduce the fraction

\frac{5}{4 y ^{5}}

\frac{\frac{5}{4 y ^{5}}}{\frac{20 y ^{2}}{y ^{2} - 4}} = 0

Divide the terms

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Evaluate

\frac{\frac{5}{4 y ^{5}}}{\frac{20 y ^{2}}{y ^{2} - 4}}

Multiply by the reciprocal

\frac{5}{4 y ^{5}} \times \frac{y ^{2} - 4}{20 y ^{2}}

Cancel out the common factor 5

\frac{1}{4 y ^{5}} \times \frac{y ^{2} - 4}{4 y ^{2}}

Multiply the terms

\frac{y ^{2} - 4}{4 y ^{5} \times 4 y ^{2}}

Multiply the terms

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Evaluate

4 y^{5} \times 4 y^{2}

Multiply the numbers

16 y^{5} \times y^{2}

Multiply the terms

16 y^{7}

\frac{y ^{2} - 4}{16 y ^{7}}

\frac{y ^{2} - 4}{16 y ^{7}} = 0

Cross multiply

y^{2} - 4 = 16 y^{7} \times 0

Simplify the equation

y^{2} - 4 = 0

Move the constant to the right side

y^{2} = 4

Take the root of both sides of the equation and remember to use both positive and negative roots

y = \pm 4

Simplify the expression

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Evaluate

4

Write the number in exponential form with the base of 2

2^{2}

Reduce the index of the radical and exponent with 2

2

y = \pm 2

Separate the equation into 2 possible cases

y = 2 y = - 2

Check if the solution is in the defined range

y = 2 y = - 2, y \in (- \infty, - 2) \cup (- 2, 0) \cup (0, 2) \cup (2, + \infty)

Solution

y \in \emptyset

Show Solution