Solve 3a^2b^5/3^4ab^5

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

Evaluate

$$\frac{a}{27}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

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

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

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

Subtract 4 from 1.

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

Subtract 1 from 2.

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

Subtract 5 from 5.

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

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

$$3^{-3}a$$

Differentiate w.r.t. a

$$\frac{1}{27} = 0.037037037037037035$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3b^{5}}{81b^{5}}a^{2-1})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\frac{1}{27}a^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{1}{27}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{1}{27}$$