Solve [(frac{6/9)^{10}(6/9)^{6}]^3}{[(6/9)^-3(6/9)^5]^-2}

$$\frac { [ ( \frac { 6 } { 9 } ) ^ { 10 } ( \frac { 6 } { 9 } ) ^ { 6 } ] ^ { 3 } } { [ ( \frac { 6 } { 9 } ) ^ { - 3 } ( \frac { 6 } { 9 } ) ^ { 5 } ] ^ { - 2 } }$$

Evaluate

$$\frac{4503599627370496}{6461081889226673298932241}\approx 6.970349091 \cdot 10^{-10}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 10 and 6 to get 16.

$$\frac{\left(\left(\frac{6}{9}\right)^{16}\right)^{3}}{\left(\left(\frac{6}{9}\right)^{-3}\times \left(\frac{6}{9}\right)^{5}\right)^{-2}}$$

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

$$\frac{\left(\frac{6}{9}\right)^{48}}{\left(\left(\frac{6}{9}\right)^{-3}\times \left(\frac{6}{9}\right)^{5}\right)^{-2}}$$

To multiply powers of the same base, add their exponents. Add -3 and 5 to get 2.

$$\frac{\left(\frac{6}{9}\right)^{48}}{\left(\left(\frac{6}{9}\right)^{2}\right)^{-2}}$$

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

$$\frac{\left(\frac{6}{9}\right)^{48}}{\left(\frac{6}{9}\right)^{-4}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -4 from 48 to get 52.

$$\left(\frac{6}{9}\right)^{52}$$

Reduce the fraction \frac{6}{9} to lowest terms by extracting and canceling out 3.

$$\left(\frac{2}{3}\right)^{52}$$

Calculate \frac{2}{3} to the power of 52 and get \frac{4503599627370496}{6461081889226673298932241}.

$$\frac{4503599627370496}{6461081889226673298932241}$$

Factor

$$\frac{2 ^ {52}}{3 ^ {52}} = 6.970349091039815 \times 10^{-10}$$