Solve (8/(27x ^ 15)) ^ (- 1/3) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$(\frac{8}{27x^{15}})^{-\frac{1}{3}}$$

Evaluate

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

Solution Steps

To raise $\frac{8}{27x^{15}}$ to a power, raise both numerator and denominator to the power and then divide.

$$\frac{8^{-\frac{1}{3}}}{\left(27x^{15}\right)^{-\frac{1}{3}}}$$

Calculate $8$ to the power of $-\frac{1}{3}$ and get $\frac{1}{2}$.

$$\frac{\frac{1}{2}}{\left(27x^{15}\right)^{-\frac{1}{3}}}$$

Expand $\left(27x^{15}\right)^{-\frac{1}{3}}$.

$$\frac{\frac{1}{2}}{27^{-\frac{1}{3}}\left(x^{15}\right)^{-\frac{1}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply $15$ and $-\frac{1}{3}$ to get $-5$.

$$\frac{\frac{1}{2}}{27^{-\frac{1}{3}}x^{-5}}$$

Calculate $27$ to the power of $-\frac{1}{3}$ and get $\frac{1}{3}$.

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

Express $\frac{\frac{1}{2}}{\frac{1}{3}x^{-5}}$ as a single fraction.

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

Multiply $2$ and $\frac{1}{3}$ to get $\frac{2}{3}$.

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

Differentiate w.r.t. x

$\frac{15x^{4}}{2}$

Steps Using Chain Rule

To raise $\frac{8}{27x^{15}}$ to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8^{-\frac{1}{3}}}{\left(27x^{15}\right)^{-\frac{1}{3}}})$$

Calculate $8$ to the power of $-\frac{1}{3}$ and get $\frac{1}{2}$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{2}}{\left(27x^{15}\right)^{-\frac{1}{3}}})$$

Expand $\left(27x^{15}\right)^{-\frac{1}{3}}$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{2}}{27^{-\frac{1}{3}}\left(x^{15}\right)^{-\frac{1}{3}}})$$

To raise a power to another power, multiply the exponents. Multiply $15$ and $-\frac{1}{3}$ to get $-5$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{2}}{27^{-\frac{1}{3}}x^{-5}})$$

Calculate $27$ to the power of $-\frac{1}{3}$ and get $\frac{1}{3}$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{2}}{\frac{1}{3}x^{-5}})$$

Express $\frac{\frac{1}{2}}{\frac{1}{3}x^{-5}}$ as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2\times \frac{1}{3}x^{-5}})$$

Multiply $2$ and $\frac{1}{3}$ to get $\frac{2}{3}$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\frac{2}{3}x^{-5}})$$

If $F$ is the composition of two differentiable functions $f\left(u\right)$ and $u=g\left(x\right)$, that is, if $F\left(x\right)=f\left(g\left(x\right)\right)$, then the derivative of $F$ is the derivative of $f$ with respect to $u$ times the derivative of $g$ with respect to $x$, that is, $\frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right)$.

$$-\left(\frac{2}{3}x^{-5}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2}{3}x^{-5})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is $0$. The derivative of $ax^{n}$ is $nax^{n-1}$.

$$-\left(\frac{2}{3}x^{-5}\right)^{-2}\left(-5\right)\times \frac{2}{3}x^{-5-1}$$

Simplify.

$$\frac{10}{3}x^{-6}\times \left(\frac{2}{3}x^{-5}\right)^{-2}$$