Solve 30a^3b/45a^4b

$$\frac { 30 a ^ { 3 } b } { 45 a ^ { 4 } b }$$

Evaluate

$$\frac{2}{3a}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{30^{1}a^{3}b^{1}}{45^{1}a^{4}b^{1}}$$

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

$$\frac{30^{1}}{45^{1}}a^{3-4}b^{1-1}$$

Subtract 4 from 3.

$$\frac{30^{1}}{45^{1}}\times \frac{1}{a}b^{1-1}$$

Subtract 1 from 1.

$$\frac{30^{1}}{45^{1}}\times \frac{1}{a}b^{0}$$

For any number a except 0, a^{0}=1.

$$\frac{30^{1}}{45^{1}}\times \frac{1}{a}$$

Reduce the fraction \frac{30}{45} to lowest terms by extracting and canceling out 15.

$$\frac{2}{3}\times \frac{1}{a}$$

Differentiate w.r.t. a

$$-\frac{2}{3a^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{30b}{45b}a^{3-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{2}{3}\times \frac{1}{a})$$

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}.

$$-\frac{2}{3}a^{-1-1}$$

Do the arithmetic.

$$-\frac{2}{3}a^{-2}$$