Question

Find the value of the following expression: (3^8•2^-5•9^0)^-2•(2^-2/3^3)^4•3^28 Write your answer in simplified form. ​

Answer 1

Answer:

4

Step-by-step explanation:

Given expression:

$(3^{8} \cdot 2^{-5} \cdot 9^{0})^{-2} \cdot \left(\dfrac{2^{-2}}{3^{3}}\right)^{4} \cdot 3^{28}$

Any number to the power of zero is 1:

$\implies (3^{8} \cdot 2^{-5} \cdot 1)^{-2} \cdot \left(\dfrac{2^{-2}}{3^{3}}\right)^{4} \cdot 3^{28}$

$\implies (3^{8} \cdot 2^{-5})^{-2} \cdot \left(\dfrac{2^{-2}}{3^{3}}\right)^{4} \cdot 3^{28}$

$\textsf{Apply the exponent rule} \quad (a^b \cdot c^d)^p=a^{bp}\cdot c^{dp}:$

$\implies 3^{(8 \cdot -2)} \cdot 2^{(-5 \cdot -2)} \cdot \left(\dfrac{2^{-2}}{3^{3}}\right)^{4} \cdot 3^{28}$

$\implies 3^{-16} \cdot 2^{10} \cdot \left(\dfrac{2^{-2}}{3^{3}}\right)^{4} \cdot 3^{28}$

$\textsf{Apply the exponent rule} \quad \left(\dfrac{a}{b}\right)^c=\dfrac{a^c}{b^c}:$

$\implies 3^{-16} \cdot 2^{10} \cdot \dfrac{\left(2^{-2}\right)^{4} }{\left(3^{3}\right)^{4} } \cdot 3^{28}$

$\textsf{Apply the exponent rule} \quad (a^b)^c=a^{bc}:$

$\implies 3^{-16} \cdot 2^{10} \cdot \dfrac{2^{(-2 \cdot 4)}}{3^{(3\cdot 4)}} \cdot 3^{28}$

$\implies 3^{-16} \cdot 2^{10} \cdot \dfrac{2^{-8}}{3^{12}} \cdot 3^{28}$

$\textsf{Apply the exponent rule} \quad \dfrac{1}{a^n}=a^{-n}$

$\implies 3^{-16} \cdot 2^{10} \cdot 2^{-8} \cdot 3^{-12} \cdot 3^{28}$

Gather like terms:

$\implies 2^{10} \cdot 2^{-8} \cdot3^{-16} \cdot 3^{-12} \cdot 3^{28}$

$\textsf{Apply the exponent rule} \quad a^b \cdot a^c=a^{b+c}:$

$\implies 2^{(10-8)}\cdot3^{(-16-12+28)}$

$\implies 2^{2}\cdot3^{0}$

Any number to the power of zero is 1:

$\implies 2^{2}\cdot 1$

$\implies 2^2$

Therefore, the solution is:

$\implies 2^2=2 \times 2=4$