Solve [(frac{a^{-2}x}{y^{-2}b})^{24}div(frac{a^{-3}x^-2}{y^-3b^-1})^16]

$$[ ( \frac { a ^ { - 2 } x } { y ^ { - 2 } b } ) ^ { 24 } \div ( \frac { a ^ { - 3 } x ^ { - 2 } } { y ^ { - 3 } b ^ { - 1 } } ) ^ { 16 } ]$$

Evaluate

$$\frac{x^{56}}{b^{40}}$$

Solution Steps

To raise \frac{a^{-2}x}{y^{-2}b} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(a^{-2}x\right)^{24}}{\left(y^{-2}b\right)^{24}}}{\left(\frac{a^{-3}x^{-2}}{y^{-3}b^{-1}}\right)^{16}}$$

To raise \frac{a^{-3}x^{-2}}{y^{-3}b^{-1}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(a^{-2}x\right)^{24}}{\left(y^{-2}b\right)^{24}}}{\frac{\left(a^{-3}x^{-2}\right)^{16}}{\left(y^{-3}b^{-1}\right)^{16}}}$$

Divide \frac{\left(a^{-2}x\right)^{24}}{\left(y^{-2}b\right)^{24}} by \frac{\left(a^{-3}x^{-2}\right)^{16}}{\left(y^{-3}b^{-1}\right)^{16}} by multiplying \frac{\left(a^{-2}x\right)^{24}}{\left(y^{-2}b\right)^{24}} by the reciprocal of \frac{\left(a^{-3}x^{-2}\right)^{16}}{\left(y^{-3}b^{-1}\right)^{16}}.

$$\frac{\left(a^{-2}x\right)^{24}\left(y^{-3}b^{-1}\right)^{16}}{\left(y^{-2}b\right)^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

Expand \left(a^{-2}x\right)^{24}.

$$\frac{\left(a^{-2}\right)^{24}x^{24}\left(y^{-3}b^{-1}\right)^{16}}{\left(y^{-2}b\right)^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and 24 to get -48.

$$\frac{a^{-48}x^{24}\left(y^{-3}b^{-1}\right)^{16}}{\left(y^{-2}b\right)^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

Expand \left(y^{-3}b^{-1}\right)^{16}.

$$\frac{a^{-48}x^{24}\left(y^{-3}\right)^{16}\left(b^{-1}\right)^{16}}{\left(y^{-2}b\right)^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 16 to get -48.

$$\frac{a^{-48}x^{24}y^{-48}\left(b^{-1}\right)^{16}}{\left(y^{-2}b\right)^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 16 to get -16.

$$\frac{a^{-48}x^{24}y^{-48}b^{-16}}{\left(y^{-2}b\right)^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

Expand \left(y^{-2}b\right)^{24}.

$$\frac{a^{-48}x^{24}y^{-48}b^{-16}}{\left(y^{-2}\right)^{24}b^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and 24 to get -48.

$$\frac{a^{-48}x^{24}y^{-48}b^{-16}}{y^{-48}b^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

Expand \left(a^{-3}x^{-2}\right)^{16}.

$$\frac{a^{-48}x^{24}y^{-48}b^{-16}}{y^{-48}b^{24}\left(a^{-3}\right)^{16}\left(x^{-2}\right)^{16}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 16 to get -48.

$$\frac{a^{-48}x^{24}y^{-48}b^{-16}}{y^{-48}b^{24}a^{-48}\left(x^{-2}\right)^{16}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and 16 to get -32.

$$\frac{a^{-48}x^{24}y^{-48}b^{-16}}{y^{-48}b^{24}a^{-48}x^{-32}}$$

Cancel out a^{-48}y^{-48} in both numerator and denominator.

$$\frac{b^{-16}x^{24}}{x^{-32}b^{24}}$$

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

$$\frac{b^{-16}x^{56}}{b^{24}}$$

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

$$\frac{x^{56}}{b^{40}}$$

Expand

$$\frac{x^{56}}{b^{40}}$$

Solution Steps

To raise \frac{a^{-2}x}{y^{-2}b} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(a^{-2}x\right)^{24}}{\left(y^{-2}b\right)^{24}}}{\left(\frac{a^{-3}x^{-2}}{y^{-3}b^{-1}}\right)^{16}}$$

To raise \frac{a^{-3}x^{-2}}{y^{-3}b^{-1}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(a^{-2}x\right)^{24}}{\left(y^{-2}b\right)^{24}}}{\frac{\left(a^{-3}x^{-2}\right)^{16}}{\left(y^{-3}b^{-1}\right)^{16}}}$$

Divide \frac{\left(a^{-2}x\right)^{24}}{\left(y^{-2}b\right)^{24}} by \frac{\left(a^{-3}x^{-2}\right)^{16}}{\left(y^{-3}b^{-1}\right)^{16}} by multiplying \frac{\left(a^{-2}x\right)^{24}}{\left(y^{-2}b\right)^{24}} by the reciprocal of \frac{\left(a^{-3}x^{-2}\right)^{16}}{\left(y^{-3}b^{-1}\right)^{16}}.

$$\frac{\left(a^{-2}x\right)^{24}\left(y^{-3}b^{-1}\right)^{16}}{\left(y^{-2}b\right)^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

Expand \left(a^{-2}x\right)^{24}.

$$\frac{\left(a^{-2}\right)^{24}x^{24}\left(y^{-3}b^{-1}\right)^{16}}{\left(y^{-2}b\right)^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and 24 to get -48.

$$\frac{a^{-48}x^{24}\left(y^{-3}b^{-1}\right)^{16}}{\left(y^{-2}b\right)^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

Expand \left(y^{-3}b^{-1}\right)^{16}.

$$\frac{a^{-48}x^{24}\left(y^{-3}\right)^{16}\left(b^{-1}\right)^{16}}{\left(y^{-2}b\right)^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 16 to get -48.

$$\frac{a^{-48}x^{24}y^{-48}\left(b^{-1}\right)^{16}}{\left(y^{-2}b\right)^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and 16 to get -16.

$$\frac{a^{-48}x^{24}y^{-48}b^{-16}}{\left(y^{-2}b\right)^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

Expand \left(y^{-2}b\right)^{24}.

$$\frac{a^{-48}x^{24}y^{-48}b^{-16}}{\left(y^{-2}\right)^{24}b^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and 24 to get -48.

$$\frac{a^{-48}x^{24}y^{-48}b^{-16}}{y^{-48}b^{24}\left(a^{-3}x^{-2}\right)^{16}}$$

Expand \left(a^{-3}x^{-2}\right)^{16}.

$$\frac{a^{-48}x^{24}y^{-48}b^{-16}}{y^{-48}b^{24}\left(a^{-3}\right)^{16}\left(x^{-2}\right)^{16}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 16 to get -48.

$$\frac{a^{-48}x^{24}y^{-48}b^{-16}}{y^{-48}b^{24}a^{-48}\left(x^{-2}\right)^{16}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and 16 to get -32.

$$\frac{a^{-48}x^{24}y^{-48}b^{-16}}{y^{-48}b^{24}a^{-48}x^{-32}}$$

Cancel out a^{-48}y^{-48} in both numerator and denominator.

$$\frac{b^{-16}x^{24}}{x^{-32}b^{24}}$$

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

$$\frac{b^{-16}x^{56}}{b^{24}}$$

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

$$\frac{x^{56}}{b^{40}}$$