Solve frac{15x^{-5}b^{8}}{35a^{7}b^{-2}}*frac{21x^{3}y^4}{9x^-2a^-3b^10}

$$\frac { 15 x ^ { - 5 } b ^ { 8 } } { 35 a ^ { 7 } b ^ { - 2 } } \cdot \frac { 21 x ^ { 3 } y ^ { 4 } } { 9 x ^ { - 2 } a ^ { - 3 } b ^ { 10 } }$$

Evaluate

$$\left(\frac{y}{a}\right)^{4}$$

Solution Steps

Cancel out 5 in both numerator and denominator.

$$\frac{3x^{-5}b^{8}}{7b^{-2}a^{7}}\times \frac{21x^{3}y^{4}}{9x^{-2}a^{-3}b^{10}}$$

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

$$\frac{3x^{-5}b^{10}}{7a^{7}}\times \frac{21x^{3}y^{4}}{9x^{-2}a^{-3}b^{10}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{3x^{-5}b^{10}}{7a^{7}}\times \frac{7x^{3}y^{4}}{3a^{-3}x^{-2}b^{10}}$$

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

$$\frac{3x^{-5}b^{10}}{7a^{7}}\times \frac{7y^{4}x^{5}}{3a^{-3}b^{10}}$$

Multiply \frac{3x^{-5}b^{10}}{7a^{7}} times \frac{7y^{4}x^{5}}{3a^{-3}b^{10}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3x^{-5}b^{10}\times 7y^{4}x^{5}}{7a^{7}\times 3a^{-3}b^{10}}$$

Cancel out 3\times 7b^{10} in both numerator and denominator.

$$\frac{x^{-5}y^{4}x^{5}}{a^{-3}a^{7}}$$

Multiply x^{-5} and x^{5} to get 1.

$$\frac{y^{4}}{a^{-3}a^{7}}$$

To multiply powers of the same base, add their exponents. Add -3 and 7 to get 4.

$$\frac{y^{4}}{a^{4}}$$

Differentiate w.r.t. a

$$-\frac{4y^{4}}{a^{5}}$$