Solve 10a^-3/12a^-5

$$\frac { 10 a ^ { - 3 } } { 12 a ^ { - 5 } }$$

Evaluate

$$\frac{5a^{2}}{6}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(10a^{-3}\right)^{1}\times \frac{1}{12a^{-5}}$$

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

$$10^{1}\left(a^{-3}\right)^{1}\times \frac{1}{12}\times \frac{1}{a^{-5}}$$

Use the Commutative Property of Multiplication.

$$10^{1}\times \frac{1}{12}\left(a^{-3}\right)^{1}\times \frac{1}{a^{-5}}$$

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

$$10^{1}\times \frac{1}{12}a^{-3}a^{-5\left(-1\right)}$$

Multiply -5 times -1.

$$10^{1}\times \frac{1}{12}a^{-3}a^{5}$$

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

$$10^{1}\times \frac{1}{12}a^{-3+5}$$

Add the exponents -3 and 5.

$$10^{1}\times \frac{1}{12}a^{2}$$

Raise 10 to the power 1.

$$10\times \frac{1}{12}a^{2}$$

Multiply 10 times \frac{1}{12}.

$$\frac{5}{6}a^{2}$$

Differentiate w.r.t. a

$$\frac{5a}{3}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{10}{12}a^{-3-\left(-5\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{5}{6}a^{2})$$

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}.

$$2\times \frac{5}{6}a^{2-1}$$

Do the arithmetic.

$$\frac{5}{3}a^{1}$$

For any term t, t^{1}=t.

$$\frac{5}{3}a$$