Solve frac{(5xy^{3}z^{-2})^2}{(4x^6yz^-3)^-1}

$$\frac { ( 5 x y ^ { 3 } z ^ { - 2 } ) ^ { 2 } } { ( 4 x ^ { 6 } y z ^ { - 3 } ) ^ { - 1 } }$$

Evaluate

$$100\times \left(\frac{y}{z}\right)^{7}x^{8}$$

Solution Steps

Expand \left(5xy^{3}z^{-2}\right)^{2}.

$$\frac{5^{2}x^{2}\left(y^{3}\right)^{2}\left(z^{-2}\right)^{2}}{\left(4x^{6}yz^{-3}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{5^{2}x^{2}y^{6}\left(z^{-2}\right)^{2}}{\left(4x^{6}yz^{-3}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and 2 to get -4.

$$\frac{5^{2}x^{2}y^{6}z^{-4}}{\left(4x^{6}yz^{-3}\right)^{-1}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{25x^{2}y^{6}z^{-4}}{\left(4x^{6}yz^{-3}\right)^{-1}}$$

Expand \left(4x^{6}yz^{-3}\right)^{-1}.

$$\frac{25x^{2}y^{6}z^{-4}}{4^{-1}\left(x^{6}\right)^{-1}y^{-1}\left(z^{-3}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -1 to get -6.

$$\frac{25x^{2}y^{6}z^{-4}}{4^{-1}x^{-6}y^{-1}\left(z^{-3}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -1 to get 3.

$$\frac{25x^{2}y^{6}z^{-4}}{4^{-1}x^{-6}y^{-1}z^{3}}$$

Calculate 4 to the power of -1 and get \frac{1}{4}.

$$\frac{25x^{2}y^{6}z^{-4}}{\frac{1}{4}x^{-6}y^{-1}z^{3}}$$

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

$$\frac{25z^{-4}y^{7}x^{8}}{\frac{1}{4}z^{3}}$$

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

$$\frac{25y^{7}x^{8}}{\frac{1}{4}z^{7}}$$

Expand

$$100\times \left(\frac{y}{z}\right)^{7}x^{8}$$

Solution Steps

Expand \left(5xy^{3}z^{-2}\right)^{2}.

$$\frac{5^{2}x^{2}\left(y^{3}\right)^{2}\left(z^{-2}\right)^{2}}{\left(4x^{6}yz^{-3}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{5^{2}x^{2}y^{6}\left(z^{-2}\right)^{2}}{\left(4x^{6}yz^{-3}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and 2 to get -4.

$$\frac{5^{2}x^{2}y^{6}z^{-4}}{\left(4x^{6}yz^{-3}\right)^{-1}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{25x^{2}y^{6}z^{-4}}{\left(4x^{6}yz^{-3}\right)^{-1}}$$

Expand \left(4x^{6}yz^{-3}\right)^{-1}.

$$\frac{25x^{2}y^{6}z^{-4}}{4^{-1}\left(x^{6}\right)^{-1}y^{-1}\left(z^{-3}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -1 to get -6.

$$\frac{25x^{2}y^{6}z^{-4}}{4^{-1}x^{-6}y^{-1}\left(z^{-3}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -1 to get 3.

$$\frac{25x^{2}y^{6}z^{-4}}{4^{-1}x^{-6}y^{-1}z^{3}}$$

Calculate 4 to the power of -1 and get \frac{1}{4}.

$$\frac{25x^{2}y^{6}z^{-4}}{\frac{1}{4}x^{-6}y^{-1}z^{3}}$$

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

$$\frac{25z^{-4}y^{7}x^{8}}{\frac{1}{4}z^{3}}$$

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

$$\frac{25y^{7}x^{8}}{\frac{1}{4}z^{7}}$$