Question

Which is the simplified form of the expression (2⁻² ⋅ 3⁴)⁻³ ⋅ ((2⁻³ ⋅ 3²)²) ?

a) (1)(216)
b) (1)(432)
c) (1)(864)
d) (1)(1728)

Answer 1

To simplify the expression (2⁻² ⋅ 3⁴)⁻³ ⋅ ((2⁻³ ⋅ 3²)²), we need to perform the calculations inside the parentheses first. Starting with the expression inside the first set of parentheses, 2⁻² ⋅ 3⁴ is equal to 81/4. Next, we simplify the expression inside the second set of parentheses, 2⁻³ ⋅ 3² which simplifies to 9/8. Finally, we substitute the simplified expressions back into the original expression and simplify to 1/6561.

Step-by-step explanation:

To simplify the expression (2⁻² ⋅ 3⁴)⁻³ ⋅ ((2⁻³ ⋅ 3²)²), we need to perform the calculations inside the parentheses first. Starting with the expression inside the first set of parentheses, 2⁻² ⋅ 3⁴ is equal to (1/2²) ⋅ (3⁴), which simplifies to (1/4) ⋅ (81), or 81/4.

Next, we simplify the expression inside the second set of parentheses, 2⁻³ ⋅ 3². This is equal to (1/2³) ⋅ (3²), which simplifies to (1/8) ⋅ 9, or 9/8.

Now, we substitute the simplified expressions back into the original expression ((1/4) ⋅ (81))^(-3) ⋅ (9/8)^2. Applying the power of -3 to (81/4) gives us (4/81)^3. Similarly, applying the power of 2 to (9/8) gives us (9/8)^2.

Finally, we multiply (4/81)^3 with (9/8)^2. This simplifies to (4^3)/(81^3) ⋅ (9^2)/(8^2), which further simplifies to 64/531441 ⋅ 81/64. The 64 cancels out, leaving us with 81/531441, or 1/6561.