Solve (frac{10^{-2}a^{-2}b^{-3}c^{2}}{5^{-3}a^{-4}b^{5}c^{-1}})^{-3}:(frac{5a^{-3}b^{5}c^{-4}}{2^{0}a^-3b^3c^-5})^-2

$$( \frac { 10 ^ { - 2 } a ^ { - 2 } b ^ { - 3 } c ^ { 2 } } { 5 ^ { - 3 } a ^ { - 4 } b ^ { 5 } c ^ { - 1 } } ) ^ { - 3 } : ( \frac { 5 a ^ { - 3 } b ^ { 5 } c ^ { - 4 } } { 2 ^ { 0 } a ^ { - 3 } b ^ { 3 } c ^ { - 5 } } ) ^ { - 2 }$$

Evaluate

$$\frac{64b^{28}}{5a^{6}c^{7}}$$

Solution Steps

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

$$\frac{\left(\frac{10^{-2}b^{-3}a^{2}c^{3}}{5^{-3}b^{5}}\right)^{-3}}{\left(\frac{5a^{-3}b^{5}c^{-4}}{2^{0}a^{-3}b^{3}c^{-5}}\right)^{-2}}$$

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

$$\frac{\left(\frac{10^{-2}a^{2}c^{3}}{5^{-3}b^{8}}\right)^{-3}}{\left(\frac{5a^{-3}b^{5}c^{-4}}{2^{0}a^{-3}b^{3}c^{-5}}\right)^{-2}}$$

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$\frac{\left(\frac{\frac{1}{100}a^{2}c^{3}}{5^{-3}b^{8}}\right)^{-3}}{\left(\frac{5a^{-3}b^{5}c^{-4}}{2^{0}a^{-3}b^{3}c^{-5}}\right)^{-2}}$$

Calculate 5 to the power of -3 and get \frac{1}{125}.

$$\frac{\left(\frac{\frac{1}{100}a^{2}c^{3}}{\frac{1}{125}b^{8}}\right)^{-3}}{\left(\frac{5a^{-3}b^{5}c^{-4}}{2^{0}a^{-3}b^{3}c^{-5}}\right)^{-2}}$$

To raise \frac{\frac{1}{100}a^{2}c^{3}}{\frac{1}{125}b^{8}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\left(\frac{5a^{-3}b^{5}c^{-4}}{2^{0}a^{-3}b^{3}c^{-5}}\right)^{-2}}$$

Cancel out a^{-3}b^{3} in both numerator and denominator.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\left(\frac{5c^{-4}b^{2}}{2^{0}c^{-5}}\right)^{-2}}$$

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

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\left(\frac{5c^{1}b^{2}}{2^{0}}\right)^{-2}}$$

Calculate c to the power of 1 and get c.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\left(\frac{5cb^{2}}{2^{0}}\right)^{-2}}$$

Calculate 2 to the power of 0 and get 1.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\left(\frac{5cb^{2}}{1}\right)^{-2}}$$

Anything divided by one gives itself.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\left(5cb^{2}\right)^{-2}}$$

Expand \left(5cb^{2}\right)^{-2}.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{5^{-2}c^{-2}\left(b^{2}\right)^{-2}}$$

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

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{5^{-2}c^{-2}b^{-4}}$$

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\frac{1}{25}c^{-2}b^{-4}}$$

Express \frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\frac{1}{25}c^{-2}b^{-4}} as a single fraction.

$$\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}\times \frac{1}{25}c^{-2}b^{-4}}$$

Expand \left(\frac{1}{100}a^{2}c^{3}\right)^{-3}.

$$\frac{\left(\frac{1}{100}\right)^{-3}\left(a^{2}\right)^{-3}\left(c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}\times \frac{1}{25}c^{-2}b^{-4}}$$

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

$$\frac{\left(\frac{1}{100}\right)^{-3}a^{-6}\left(c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}\times \frac{1}{25}c^{-2}b^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -3 to get -9.

$$\frac{\left(\frac{1}{100}\right)^{-3}a^{-6}c^{-9}}{\left(\frac{1}{125}b^{8}\right)^{-3}\times \frac{1}{25}c^{-2}b^{-4}}$$

Calculate \frac{1}{100} to the power of -3 and get 1000000.

$$\frac{1000000a^{-6}c^{-9}}{\left(\frac{1}{125}b^{8}\right)^{-3}\times \frac{1}{25}c^{-2}b^{-4}}$$

Expand \left(\frac{1}{125}b^{8}\right)^{-3}.

$$\frac{1000000a^{-6}c^{-9}}{\left(\frac{1}{125}\right)^{-3}\left(b^{8}\right)^{-3}\times \frac{1}{25}c^{-2}b^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and -3 to get -24.

$$\frac{1000000a^{-6}c^{-9}}{\left(\frac{1}{125}\right)^{-3}b^{-24}\times \frac{1}{25}c^{-2}b^{-4}}$$

Calculate \frac{1}{125} to the power of -3 and get 1953125.

$$\frac{1000000a^{-6}c^{-9}}{1953125b^{-24}\times \frac{1}{25}c^{-2}b^{-4}}$$

Multiply 1953125 and \frac{1}{25} to get 78125.

$$\frac{1000000a^{-6}c^{-9}}{78125b^{-24}c^{-2}b^{-4}}$$

To multiply powers of the same base, add their exponents. Add -24 and -4 to get -28.

$$\frac{1000000a^{-6}c^{-9}}{78125b^{-28}c^{-2}}$$

Cancel out 15625 in both numerator and denominator.

$$\frac{64c^{-9}a^{-6}}{5b^{-28}c^{-2}}$$

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

$$\frac{64a^{-6}}{5b^{-28}c^{7}}$$

Expand

$$\frac{64b^{28}}{5a^{6}c^{7}}$$

Solution Steps

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

$$\frac{\left(\frac{10^{-2}b^{-3}a^{2}c^{3}}{5^{-3}b^{5}}\right)^{-3}}{\left(\frac{5a^{-3}b^{5}c^{-4}}{2^{0}a^{-3}b^{3}c^{-5}}\right)^{-2}}$$

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

$$\frac{\left(\frac{10^{-2}a^{2}c^{3}}{5^{-3}b^{8}}\right)^{-3}}{\left(\frac{5a^{-3}b^{5}c^{-4}}{2^{0}a^{-3}b^{3}c^{-5}}\right)^{-2}}$$

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$\frac{\left(\frac{\frac{1}{100}a^{2}c^{3}}{5^{-3}b^{8}}\right)^{-3}}{\left(\frac{5a^{-3}b^{5}c^{-4}}{2^{0}a^{-3}b^{3}c^{-5}}\right)^{-2}}$$

Calculate 5 to the power of -3 and get \frac{1}{125}.

$$\frac{\left(\frac{\frac{1}{100}a^{2}c^{3}}{\frac{1}{125}b^{8}}\right)^{-3}}{\left(\frac{5a^{-3}b^{5}c^{-4}}{2^{0}a^{-3}b^{3}c^{-5}}\right)^{-2}}$$

To raise \frac{\frac{1}{100}a^{2}c^{3}}{\frac{1}{125}b^{8}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\left(\frac{5a^{-3}b^{5}c^{-4}}{2^{0}a^{-3}b^{3}c^{-5}}\right)^{-2}}$$

Cancel out a^{-3}b^{3} in both numerator and denominator.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\left(\frac{5c^{-4}b^{2}}{2^{0}c^{-5}}\right)^{-2}}$$

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

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\left(\frac{5c^{1}b^{2}}{2^{0}}\right)^{-2}}$$

Calculate c to the power of 1 and get c.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\left(\frac{5cb^{2}}{2^{0}}\right)^{-2}}$$

Calculate 2 to the power of 0 and get 1.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\left(\frac{5cb^{2}}{1}\right)^{-2}}$$

Anything divided by one gives itself.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\left(5cb^{2}\right)^{-2}}$$

Expand \left(5cb^{2}\right)^{-2}.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{5^{-2}c^{-2}\left(b^{2}\right)^{-2}}$$

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

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{5^{-2}c^{-2}b^{-4}}$$

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\frac{1}{25}c^{-2}b^{-4}}$$

Express \frac{\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}}}{\frac{1}{25}c^{-2}b^{-4}} as a single fraction.

$$\frac{\left(\frac{1}{100}a^{2}c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}\times \frac{1}{25}c^{-2}b^{-4}}$$

Expand \left(\frac{1}{100}a^{2}c^{3}\right)^{-3}.

$$\frac{\left(\frac{1}{100}\right)^{-3}\left(a^{2}\right)^{-3}\left(c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}\times \frac{1}{25}c^{-2}b^{-4}}$$

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

$$\frac{\left(\frac{1}{100}\right)^{-3}a^{-6}\left(c^{3}\right)^{-3}}{\left(\frac{1}{125}b^{8}\right)^{-3}\times \frac{1}{25}c^{-2}b^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -3 to get -9.

$$\frac{\left(\frac{1}{100}\right)^{-3}a^{-6}c^{-9}}{\left(\frac{1}{125}b^{8}\right)^{-3}\times \frac{1}{25}c^{-2}b^{-4}}$$

Calculate \frac{1}{100} to the power of -3 and get 1000000.

$$\frac{1000000a^{-6}c^{-9}}{\left(\frac{1}{125}b^{8}\right)^{-3}\times \frac{1}{25}c^{-2}b^{-4}}$$

Expand \left(\frac{1}{125}b^{8}\right)^{-3}.

$$\frac{1000000a^{-6}c^{-9}}{\left(\frac{1}{125}\right)^{-3}\left(b^{8}\right)^{-3}\times \frac{1}{25}c^{-2}b^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and -3 to get -24.

$$\frac{1000000a^{-6}c^{-9}}{\left(\frac{1}{125}\right)^{-3}b^{-24}\times \frac{1}{25}c^{-2}b^{-4}}$$

Calculate \frac{1}{125} to the power of -3 and get 1953125.

$$\frac{1000000a^{-6}c^{-9}}{1953125b^{-24}\times \frac{1}{25}c^{-2}b^{-4}}$$

Multiply 1953125 and \frac{1}{25} to get 78125.

$$\frac{1000000a^{-6}c^{-9}}{78125b^{-24}c^{-2}b^{-4}}$$

To multiply powers of the same base, add their exponents. Add -24 and -4 to get -28.

$$\frac{1000000a^{-6}c^{-9}}{78125b^{-28}c^{-2}}$$

Cancel out 15625 in both numerator and denominator.

$$\frac{64c^{-9}a^{-6}}{5b^{-28}c^{-2}}$$

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

$$\frac{64a^{-6}}{5b^{-28}c^{7}}$$