Solve h(t)=-4t^2 8t^32

Question

Function

Find the inverse

Evaluate the derivative

Find the domain

h^{- 1} (t) = - \frac{5 16 t}{4}

Evaluate

h (t) = - 4 t^{2} \times 8 t^{3} \times 2

Simplify

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Evaluate

- 4 t^{2} \times 8 t^{3} \times 2

Multiply the terms

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Evaluate

4 \times 8 \times 2

Multiply the terms

32 \times 2

Multiply the numbers

64

- 64 t^{2} \times t^{3}

Multiply the terms with the same base by adding their exponents

- 64 t^{2 + 3}

Add the numbers

- 64 t^{5}

h (t) = - 64 t^{5}

In the equation for h (t),replace h (t) with y

y = - 64 t^{5}

Interchange t and y

t = - 64 y^{5}

Swap the sides of the equation

- 64 y^{5} = t

Change the signs on both sides of the equation

64 y^{5} = - t

Divide both sides

\frac{64 y ^{5}}{64} = \frac{- t}{64}

Divide the numbers

y^{5} = \frac{- t}{64}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

y^{5} = - \frac{t}{64}

Take the 5 -th root on both sides of the equation

5 y^{5} = 5 - \frac{t}{64}

Calculate

y = 5 - \frac{t}{64}

Simplify the root

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Evaluate

5 - \frac{t}{64}

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

\frac{5 - t}{5 64}

Simplify the radical expression

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Evaluate

564

Write the expression as a product where the root of one of the factors can be evaluated

5 32 \times 2

Write the number in exponential form with the base of 2

5 2^{5} \times 2

The root of a product is equal to the product of the roots of each factor

5 2^{5} \times 52

Reduce the index of the radical and exponent with 5

252

\frac{5 - t}{2 5 2}

Multiply by the Conjugate

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

Calculate

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

Calculate

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Evaluate

5 - t \times 5 2^{4}

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

5 - t \times 2^{4}

Calculate the product

5 - 2^{4} t

An odd root of a negative radicand is always a negative

- 5 2^{4} t

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

Calculate

\frac{- 5 2 ^{4} t}{4}

Calculate

- \frac{5 2 ^{4} t}{4}

Calculate

- \frac{5 16 t}{4}

y = - \frac{5 16 t}{4}

Solution

h^{- 1} (t) = - \frac{5 16 t}{4}

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