Solve frac{4^{5}*a^{8}b^3}{4^3*a^3b^2}

$$\frac { 4 ^ { 5 } \times a ^ { 8 } b ^ { 3 } } { 4 ^ { 3 } \times a ^ { 3 } b ^ { 2 } }$$

Evaluate

$$16ba^{5}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{4^{5}a^{8}b^{3}}{4^{3}a^{3}b^{2}}$$

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

$$4^{5-3}a^{8-3}b^{3-2}$$

Subtract 3 from 5.

$$4^{2}a^{8-3}b^{3-2}$$

Subtract 3 from 8.

$$16a^{5}b^{3-2}$$

Subtract 2 from 3.

$$16a^{5}b^{1}$$

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

$$16a^{5}b$$

Differentiate w.r.t. a

$$80ba^{4}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1024b^{3}}{64b^{2}}a^{8-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(16ba^{5})$$

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}.

$$5\times 16ba^{5-1}$$

Do the arithmetic.

$$80ba^{4}$$