Solve frac{3a^{2}b^5}{3^4a^5b^5}

$$\frac { 3 a ^ { 2 } b ^ { 5 } } { 3 ^ { 4 } a ^ { 5 } b ^ { 5 } }$$

Evaluate

$$\frac{1}{27a^{3}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{3^{1}a^{2}b^{5}}{3^{4}a^{5}b^{5}}$$

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

$$3^{1-4}a^{2-5}b^{5-5}$$

Subtract 4 from 1.

$$3^{-3}a^{2-5}b^{5-5}$$

Subtract 5 from 2.

$$\frac{1}{27}a^{-3}b^{5-5}$$

Subtract 5 from 5.

$$\frac{1}{27}a^{-3}b^{0}$$

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

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

Differentiate w.r.t. a

$$-\frac{1}{9a^{4}}$$

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{27}a^{-3})$$

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}.

$$-3\times \frac{1}{27}a^{-3-1}$$

Do the arithmetic.

$$-\frac{1}{9}a^{-4}$$