Calculus Examples

$\frac{- 2 x^{- \frac{5}{3}}}{9} = 0$

Step 1

Set the numerator equal to zero.

$- 2 x^{- \frac{5}{3}} = 0$

Step 2

Solve the equation for

$x$ .

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Step 2.1

Raise each side of the equation to the power of

$- \frac{3}{5}$ to eliminate the fractional exponent on the left side.

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

Step 2.2

Simplify the exponent.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify

${(- 2 x^{- \frac{5}{3}})}^{- \frac{3}{5}}$ .

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Step 2.2.1.1.1

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

${(- 2 \frac{1}{x^{\frac{5}{3}}})}^{- \frac{3}{5}} = 0^{- \frac{3}{5}}$

Step 2.2.1.1.2

Combine

$- 2$ and

$\frac{1}{x^{\frac{5}{3}}}$ .

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

Step 2.2.1.1.3

Move the negative in front of the fraction.

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

Step 2.2.1.1.4

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 2.2.1.1.5

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.2.1.1.5.1

Apply the product rule to

$- \frac{x^{\frac{5}{3}}}{2}$ .

${(- 1)}^{\frac{3}{5}} {(\frac{x^{\frac{5}{3}}}{2})}^{\frac{3}{5}} = 0^{- \frac{3}{5}}$

Step 2.2.1.1.5.2

Apply the product rule to

$\frac{x^{\frac{5}{3}}}{2}$ .

${(- 1)}^{\frac{3}{5}} \frac{{(x^{\frac{5}{3}})}^{\frac{3}{5}}}{2^{\frac{3}{5}}} = 0^{- \frac{3}{5}}$

Step 2.2.1.1.6

Reduce the expression by cancelling the common factors.

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Step 2.2.1.1.6.1

Rewrite

$- 1$ as

${(- 1)}^{5}$ .

${({(- 1)}^{5})}^{\frac{3}{5}} \frac{{(x^{\frac{5}{3}})}^{\frac{3}{5}}}{2^{\frac{3}{5}}} = 0^{- \frac{3}{5}}$

Step 2.2.1.1.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

${(- 1)}^{5 (\frac{3}{5})} \frac{{(x^{\frac{5}{3}})}^{\frac{3}{5}}}{2^{\frac{3}{5}}} = 0^{- \frac{3}{5}}$

Step 2.2.1.1.6.3

Cancel the common factor of

$5$ .

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Step 2.2.1.1.6.3.1

Cancel the common factor.

${(- 1)}^{5 (\frac{3}{5})} \frac{{(x^{\frac{5}{3}})}^{\frac{3}{5}}}{2^{\frac{3}{5}}} = 0^{- \frac{3}{5}}$

Step 2.2.1.1.6.3.2

Rewrite the expression.

${(- 1)}^{3} \frac{{(x^{\frac{5}{3}})}^{\frac{3}{5}}}{2^{\frac{3}{5}}} = 0^{- \frac{3}{5}}$

Step 2.2.1.1.6.4

Raise

$- 1$ to the power of

$3$ .

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

Step 2.2.1.1.7

Simplify the numerator.

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Step 2.2.1.1.7.1

Multiply the exponents in

${(x^{\frac{5}{3}})}^{\frac{3}{5}}$ .

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Step 2.2.1.1.7.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$- \frac{x^{\frac{5}{3} \cdot \frac{3}{5}}}{2^{\frac{3}{5}}} = 0^{- \frac{3}{5}}$

Step 2.2.1.1.7.1.2

Cancel the common factor of

$5$ .

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Step 2.2.1.1.7.1.2.1

Cancel the common factor.

$- \frac{x^{\frac{5}{3} \cdot \frac{3}{5}}}{2^{\frac{3}{5}}} = 0^{- \frac{3}{5}}$

Step 2.2.1.1.7.1.2.2

Rewrite the expression.

$- \frac{x^{\frac{1}{3} \cdot 3}}{2^{\frac{3}{5}}} = 0^{- \frac{3}{5}}$

Step 2.2.1.1.7.1.3

Cancel the common factor of

$3$ .

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Step 2.2.1.1.7.1.3.1

Cancel the common factor.

$- \frac{x^{\frac{1}{3} \cdot 3}}{2^{\frac{3}{5}}} = 0^{- \frac{3}{5}}$

Step 2.2.1.1.7.1.3.2

Rewrite the expression.

$- \frac{x^{1}}{2^{\frac{3}{5}}} = 0^{- \frac{3}{5}}$

Step 2.2.1.1.7.2

Simplify.

$- \frac{x}{2^{\frac{3}{5}}} = 0^{- \frac{3}{5}}$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify

$0^{- \frac{3}{5}}$ .

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Step 2.2.2.1.1

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{x}{2^{\frac{3}{5}}} = \frac{1}{0^{\frac{3}{5}}}$

Step 2.2.2.1.2

Simplify the denominator.

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Step 2.2.2.1.2.1

Rewrite

$0$ as

$0^{5}$ .

$- \frac{x}{2^{\frac{3}{5}}} = \frac{1}{{(0^{5})}^{\frac{3}{5}}}$

Step 2.2.2.1.2.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$- \frac{x}{2^{\frac{3}{5}}} = \frac{1}{0^{5 (\frac{3}{5})}}$

Step 2.2.2.1.2.3

Cancel the common factor of

$5$ .

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Step 2.2.2.1.2.3.1

Cancel the common factor.

$- \frac{x}{2^{\frac{3}{5}}} = \frac{1}{0^{5 (\frac{3}{5})}}$

Step 2.2.2.1.2.3.2

Rewrite the expression.

$- \frac{x}{2^{\frac{3}{5}}} = \frac{1}{0^{3}}$

Step 2.2.2.1.2.4

Raising

$0$ to any positive power yields

$0$ .

$- \frac{x}{2^{\frac{3}{5}}} = \frac{1}{0}$

Step 2.2.2.2

The equation cannot be solved because it is undefined.

$Undefined$