Solve 267{[(+3/4)^{2}]^{2}}^{3}*[(-3/4)^{5}*(-3/4)^{2}]^{2}*{[(+3/4)^2]^3*(-frac{3}{4})^7}^-2

$$267 \{ [ ( + \frac { 3 } { 4 } ) ^ { 2 } ] ^ { 2 } \} ^ { 3 } \cdot [ ( - \frac { 3 } { 4 } ) ^ { 5 } \cdot ( - \frac { 3 } { 4 } ) ^ { 2 } ] ^ { 2 } \cdot \{ [ ( + \frac { 3 } { 4 } ) ^ { 2 } ] ^ { 3 } \cdot ( - \frac { 3 } { 4 } ) ^ { 7 } \} ^ { - 2 }$$

Evaluate

$$267$$

Solution Steps

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

$$267\times \left(\left(\frac{3}{4}\right)^{4}\right)^{3}\left(\left(-\frac{3}{4}\right)^{5}\left(-\frac{3}{4}\right)^{2}\right)^{2}\times \left(\left(\left(\frac{3}{4}\right)^{2}\right)^{3}\left(-\frac{3}{4}\right)^{7}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$267\times \left(\frac{3}{4}\right)^{12}\left(\left(-\frac{3}{4}\right)^{5}\left(-\frac{3}{4}\right)^{2}\right)^{2}\times \left(\left(\left(\frac{3}{4}\right)^{2}\right)^{3}\left(-\frac{3}{4}\right)^{7}\right)^{-2}$$

To multiply powers of the same base, add their exponents. Add 5 and 2 to get 7.

$$267\times \left(\frac{3}{4}\right)^{12}\left(\left(-\frac{3}{4}\right)^{7}\right)^{2}\times \left(\left(\left(\frac{3}{4}\right)^{2}\right)^{3}\left(-\frac{3}{4}\right)^{7}\right)^{-2}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 2 to get 14.

$$267\times \left(\frac{3}{4}\right)^{12}\left(-\frac{3}{4}\right)^{14}\times \left(\left(\left(\frac{3}{4}\right)^{2}\right)^{3}\left(-\frac{3}{4}\right)^{7}\right)^{-2}$$

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

$$267\times \left(\frac{3}{4}\right)^{12}\left(-\frac{3}{4}\right)^{14}\times \left(\left(\frac{3}{4}\right)^{6}\left(-\frac{3}{4}\right)^{7}\right)^{-2}$$

Calculate \frac{3}{4} to the power of 12 and get \frac{531441}{16777216}.

$$267\times \frac{531441}{16777216}\left(-\frac{3}{4}\right)^{14}\times \left(\left(\frac{3}{4}\right)^{6}\left(-\frac{3}{4}\right)^{7}\right)^{-2}$$

Multiply 267 and \frac{531441}{16777216} to get \frac{141894747}{16777216}.

$$\frac{141894747}{16777216}\left(-\frac{3}{4}\right)^{14}\times \left(\left(\frac{3}{4}\right)^{6}\left(-\frac{3}{4}\right)^{7}\right)^{-2}$$

Calculate -\frac{3}{4} to the power of 14 and get \frac{4782969}{268435456}.

$$\frac{141894747}{16777216}\times \frac{4782969}{268435456}\times \left(\left(\frac{3}{4}\right)^{6}\left(-\frac{3}{4}\right)^{7}\right)^{-2}$$

Multiply \frac{141894747}{16777216} and \frac{4782969}{268435456} to get \frac{678678176163843}{4503599627370496}.

$$\frac{678678176163843}{4503599627370496}\times \left(\left(\frac{3}{4}\right)^{6}\left(-\frac{3}{4}\right)^{7}\right)^{-2}$$

Calculate \frac{3}{4} to the power of 6 and get \frac{729}{4096}.

$$\frac{678678176163843}{4503599627370496}\times \left(\frac{729}{4096}\left(-\frac{3}{4}\right)^{7}\right)^{-2}$$

Calculate -\frac{3}{4} to the power of 7 and get -\frac{2187}{16384}.

$$\frac{678678176163843}{4503599627370496}\times \left(\frac{729}{4096}\left(-\frac{2187}{16384}\right)\right)^{-2}$$

Multiply \frac{729}{4096} and -\frac{2187}{16384} to get -\frac{1594323}{67108864}.

$$\frac{678678176163843}{4503599627370496}\left(-\frac{1594323}{67108864}\right)^{-2}$$

Calculate -\frac{1594323}{67108864} to the power of -2 and get \frac{4503599627370496}{2541865828329}.

$$\frac{678678176163843}{4503599627370496}\times \frac{4503599627370496}{2541865828329}$$

Multiply \frac{678678176163843}{4503599627370496} and \frac{4503599627370496}{2541865828329} to get 267.

$$267$$

Factor

$$3\times 89$$