Solve (a^-7diva^15)div(a^-6)^4/a

$$\frac { ( a ^ { - 7 } \div a ^ { 15 } ) \div ( a ^ { - 6 } ) ^ { 4 } } { a }$$

Evaluate

$$a$$

Solution Steps

Rewrite a^{15} as a^{-7}a^{22}. Cancel out a^{-7} in both numerator and denominator.

$$\frac{\frac{\frac{1}{a^{22}}}{\left(a^{-6}\right)^{4}}}{a}$$

To raise a power to another power, multiply the exponents. Multiply -6 and 4 to get -24.

$$\frac{\frac{\frac{1}{a^{22}}}{a^{-24}}}{a}$$

Express \frac{\frac{\frac{1}{a^{22}}}{a^{-24}}}{a} as a single fraction.

$$\frac{\frac{1}{a^{22}}}{a^{-24}a}$$

To multiply powers of the same base, add their exponents. Add -24 and 1 to get -23.

$$\frac{\frac{1}{a^{22}}}{a^{-23}}$$

Express \frac{\frac{1}{a^{22}}}{a^{-23}} as a single fraction.

$$\frac{1}{a^{22}a^{-23}}$$

To multiply powers of the same base, add their exponents. Add 22 and -23 to get -1.

$$\frac{1}{a^{-1}}$$

Differentiate w.r.t. a

$$1$$

Steps Using Chain Rule

Rewrite a^{15} as a^{-7}a^{22}. Cancel out a^{-7} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{\frac{1}{a^{22}}}{\left(a^{-6}\right)^{4}}}{a})$$

To raise a power to another power, multiply the exponents. Multiply -6 and 4 to get -24.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{\frac{1}{a^{22}}}{a^{-24}}}{a})$$

Express \frac{\frac{\frac{1}{a^{22}}}{a^{-24}}}{a} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{1}{a^{22}}}{a^{-24}a})$$

To multiply powers of the same base, add their exponents. Add -24 and 1 to get -23.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{1}{a^{22}}}{a^{-23}})$$

Express \frac{\frac{1}{a^{22}}}{a^{-23}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{22}a^{-23}})$$

To multiply powers of the same base, add their exponents. Add 22 and -23 to get -1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{-1}})$$

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).

$$-\left(\frac{1}{a}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a})$$

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}.

$$-\left(\frac{1}{a}\right)^{-2}\left(-1\right)a^{-1-1}$$

Simplify.

$$a^{-2}\times \left(\frac{1}{a}\right)^{-2}$$