Solve 54a^5b^2/81a^3b^3

$$\frac { 54 a ^ { 5 } b ^ { 2 } } { 81 a ^ { 3 } b ^ { 3 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{54^{1}a^{5}b^{2}}{81^{1}a^{3}b^{3}}$$

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

$$\frac{54^{1}}{81^{1}}a^{5-3}b^{2-3}$$

Subtract 3 from 5.

$$\frac{54^{1}}{81^{1}}a^{2}b^{2-3}$$

Subtract 3 from 2.

$$\frac{54^{1}}{81^{1}}a^{2}\times \frac{1}{b}$$

Reduce the fraction \frac{54}{81} to lowest terms by extracting and canceling out 27.

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

Differentiate w.r.t. a

$$\frac{4a}{3b}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{54b^{2}}{81b^{3}}a^{5-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{2}{3b}a^{2})$$

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}.

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

Do the arithmetic.

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

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

$$\frac{4}{3b}a$$