Solve -10a^6/40a^13

$$\frac { - 10 a ^ { 6 } } { 40 a ^ { 13 } }$$

Evaluate

$$-\frac{1}{4a^{7}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(-10a^{6}\right)^{1}\times \frac{1}{40a^{13}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$\left(-10\right)^{1}\left(a^{6}\right)^{1}\times \frac{1}{40}\times \frac{1}{a^{13}}$$

Use the Commutative Property of Multiplication.

$$\left(-10\right)^{1}\times \frac{1}{40}\left(a^{6}\right)^{1}\times \frac{1}{a^{13}}$$

To raise a power to another power, multiply the exponents.

$$\left(-10\right)^{1}\times \frac{1}{40}a^{6}a^{13\left(-1\right)}$$

Multiply 13 times -1.

$$\left(-10\right)^{1}\times \frac{1}{40}a^{6}a^{-13}$$

To multiply powers of the same base, add their exponents.

$$\left(-10\right)^{1}\times \frac{1}{40}a^{6-13}$$

Add the exponents 6 and -13.

$$\left(-10\right)^{1}\times \frac{1}{40}a^{-7}$$

Raise -10 to the power 1.

$$-10\times \frac{1}{40}a^{-7}$$

Multiply -10 times \frac{1}{40}.

$$-\frac{1}{4}a^{-7}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{\left(-10\right)^{1}a^{6}}{40^{1}a^{13}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\left(-10\right)^{1}a^{6-13}}{40^{1}}$$

Subtract 13 from 6.

$$\frac{\left(-10\right)^{1}a^{-7}}{40^{1}}$$

Reduce the fraction \frac{-10}{40} to lowest terms by extracting and canceling out 10.

$$-\frac{1}{4}a^{-7}$$

Differentiate w.r.t. a

$$\frac{7}{4a^{8}}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(-\frac{10}{40}\right)a^{6-13})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(-\frac{1}{4}a^{-7})$$

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}.

$$-7\left(-\frac{1}{4}\right)a^{-7-1}$$

Do the arithmetic.

$$\frac{7}{4}a^{-8}$$