Solve (y^(2) 2y)^(2)-1

Question

Simplify the expression

4 y^{6} - 1

Evaluate

(y^{2} \times 2 y)^{2} - 1

Multiply

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Multiply the terms

y^{2} \times 2 y

Multiply the terms with the same base by adding their exponents

y^{2 + 1} \times 2

Add the numbers

y^{3} \times 2

Use the commutative property to reorder the terms

2 y^{3}

(2 y^{3})^{2} - 1

Solution

4 y^{6} - 1

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Factor the expression

(2 y^{3} - 1) (2 y^{3} + 1)

Evaluate

(y^{2} \times 2 y)^{2} - 1

Evaluate

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Evaluate

(y^{2} \times 2 y)^{2}

Multiply

More Steps

Multiply the terms

y^{2} \times 2 y

Multiply the terms with the same base by adding their exponents

y^{2 + 1} \times 2

Add the numbers

y^{3} \times 2

Use the commutative property to reorder the terms

2 y^{3}

(2 y^{3})^{2}

To raise a product to a power,raise each factor to that power

2^{2} (y^{3})^{2}

Evaluate the power

4 (y^{3})^{2}

Evaluate the power

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Evaluate

(y^{3})^{2}

Multiply the exponents

y^{3 \times 2}

Multiply the terms

y^{6}

4 y^{6}

4 y^{6} - 1

Rewrite the expression in exponential form

(2 y^{3})^{2} - 1^{2}

Solution

(2 y^{3} - 1) (2 y^{3} + 1)

Show Solution

Find the roots

y_{1} = - \frac{3 4}{2}, y_{2} = \frac{3 4}{2}

Alternative Form

y_{1} \approx - 0.793701, y_{2} \approx 0.793701

Evaluate

(y^{2} \times 2 y)^{2} - 1

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

(y^{2} \times 2 y)^{2} - 1 = 0

Multiply

More Steps

Multiply the terms

y^{2} \times 2 y

Multiply the terms with the same base by adding their exponents

y^{2 + 1} \times 2

Add the numbers

y^{3} \times 2

Use the commutative property to reorder the terms

2 y^{3}

(2 y^{3})^{2} - 1 = 0

Rewrite the expression

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Simplify

(2 y^{3})^{2} - 1

Rewrite the expression

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Evaluate

(2 y^{3})^{2}

To raise a product to a power,raise each factor to that power

2^{2} (y^{3})^{2}

Evaluate the power

4 (y^{3})^{2}

Evaluate the power

4 y^{6}

4 y^{6} - 1

4 y^{6} - 1 = 0

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

4 y^{6} = 0 + 1

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

4 y^{6} = 1

Divide both sides

\frac{4 y ^{6}}{4} = \frac{1}{4}

Divide the numbers

y^{6} = \frac{1}{4}

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

y = \pm 6 \frac{1}{4}

Simplify the expression

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Evaluate

6 \frac{1}{4}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{6 1}{6 4}

Simplify the radical expression

\frac{1}{6 4}

Simplify the radical expression

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Evaluate

64

Write the number in exponential form with the base of 2

6 2^{2}

Reduce the index of the radical and exponent with 2

32

\frac{1}{3 2}

Multiply by the Conjugate

\frac{3 2 ^{2}}{3 2 \times 3 2 ^{2}}

Simplify

\frac{3 4}{3 2 \times 3 2 ^{2}}

Multiply the numbers

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Evaluate

32 \times 3 2^{2}

The product of roots with the same index is equal to the root of the product

3 2 \times 2^{2}

Calculate the product

3 2^{3}

Reduce the index of the radical and exponent with 3

2

\frac{3 4}{2}

y = \pm \frac{3 4}{2}

Separate the equation into 2 possible cases

y = \frac{3 4}{2} y = - \frac{3 4}{2}

Solution

y_{1} = - \frac{3 4}{2}, y_{2} = \frac{3 4}{2}

Alternative Form

y_{1} \approx - 0.793701, y_{2} \approx 0.793701

Show Solution