Solve frac{(2a^{-5}b^{6})^3}{2^4a^8b^-7}

$$\frac { ( 2 a ^ { - 5 } b ^ { 6 } ) ^ { 3 } } { 2 ^ { 4 } a ^ { 8 } b ^ { - 7 } }$$

Evaluate

$$\frac{b^{25}}{2a^{23}}$$

Solution Steps

Expand \left(2a^{-5}b^{6}\right)^{3}.

$$\frac{2^{3}\left(a^{-5}\right)^{3}\left(b^{6}\right)^{3}}{2^{4}a^{8}b^{-7}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and 3 to get -15.

$$\frac{2^{3}a^{-15}\left(b^{6}\right)^{3}}{2^{4}a^{8}b^{-7}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 3 to get 18.

$$\frac{2^{3}a^{-15}b^{18}}{2^{4}a^{8}b^{-7}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{8a^{-15}b^{18}}{2^{4}a^{8}b^{-7}}$$

Calculate 2 to the power of 4 and get 16.

$$\frac{8a^{-15}b^{18}}{16a^{8}b^{-7}}$$

Cancel out 8 in both numerator and denominator.

$$\frac{a^{-15}b^{18}}{2b^{-7}a^{8}}$$

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

$$\frac{a^{-15}b^{25}}{2a^{8}}$$

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

$$\frac{b^{25}}{2a^{23}}$$

Expand

$$\frac{b^{25}}{2a^{23}}$$

Solution Steps

Expand \left(2a^{-5}b^{6}\right)^{3}.

$$\frac{2^{3}\left(a^{-5}\right)^{3}\left(b^{6}\right)^{3}}{2^{4}a^{8}b^{-7}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and 3 to get -15.

$$\frac{2^{3}a^{-15}\left(b^{6}\right)^{3}}{2^{4}a^{8}b^{-7}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 3 to get 18.

$$\frac{2^{3}a^{-15}b^{18}}{2^{4}a^{8}b^{-7}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{8a^{-15}b^{18}}{2^{4}a^{8}b^{-7}}$$

Calculate 2 to the power of 4 and get 16.

$$\frac{8a^{-15}b^{18}}{16a^{8}b^{-7}}$$

Cancel out 8 in both numerator and denominator.

$$\frac{a^{-15}b^{18}}{2b^{-7}a^{8}}$$

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

$$\frac{a^{-15}b^{25}}{2a^{8}}$$

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

$$\frac{b^{25}}{2a^{23}}$$