(16/81)^-3/4×512^2/3
(16/81)^(-3/4) = (3/2)^3 = 27/8
512^2/3 = (2^9)^(2/3) = 2^6 = 64
so,
(16/81)^-3/4 × 512^2/3 = 27/8 * 64 = 216
Apologies for the mistake in my previous response. You are correct.
(16/81)^(-3/4) = (3/2)^3 = 27/8
512^(2/3) = (2^9)^(2/3) = 2^6 = 64
So, (16/81)^-3/4 * 512^2/3 = (27/8) * 64 = 216.
To simplify the expression (16/81)^-3/4 × 512^2/3, we will simplify each part step by step.
Step 1: Simplify (16/81)^-3/4
To simplify this, we will first simplify the fractional exponent -3/4.
Recall that a negative exponent on a base can be converted to a positive exponent by taking its reciprocal.
Therefore, (-3/4)^-1 = 1/(-3/4) = -4/3.
Now, let's simplify the base (16/81) raised to the exponent -4/3.
To convert a fraction exponent to a radical form, we can rewrite it as follows:
(a/b)^m = ∛(a^m) / ∛(b^m)
So, with that in mind, we can rewrite (16/81)^-4/3 as:
∛(16/81)^-4
Now, let's calculate the value inside the cube root.
16/81 raised to the power of -4 can be rewritten as 81^4/16^4.
So, we have:
∛(81^4/16^4).
Simplifying the numerator and denominator separately, we get:
∛(531441/256).
Step 2: Simplify 512^2/3
To simplify this expression, let's calculate the cube root of 512 first.
The cube root of 512 is 8 because 8 × 8 × 8 = 512.
Now we can rewrite 512^2/3 as (8^3)^2/3.
Using the property of exponentiation, (a^m)^n = a^(m×n), we have:
(8^3)^2/3 = 8^(3×2/3) = 8^2 = 64.
Step 3: Multiply the two results together.
We have ∛(531441/256) × 64.
To multiply the two expressions, we multiply the values inside the cube root by 64.
So, ∛(531441/256) × 64 = ∛(531441 × 64)/∛256.
Simplifying the numerator and denominator separately, we get:
∛(34012224)/4 = 64∛(34012224)/4.
Therefore, the simplified expression (16/81)^-3/4 × 512^2/3 is 64∛(34012224)/4.
To simplify this expression, we need to evaluate each part separately and then multiply the results.
First, let's simplify (16/81)^(-3/4):
(16/81)^(-3/4) = (81/16)^(3/4) [since a^(-n) = 1/a^n]
Now, let's simplify 512^(2/3):
512^(2/3) = (8^3)^(2/3) [since 512 = 8^3]
Next, let's simplify (81/16)^(3/4):
(81/16)^(3/4) = (3^4)/(2^4)^(3/4) [since 81 = 3^4 and 16 = 2^4]
(3^4)/(2^4)^(3/4) = (3^4)/(2^3) [since (2^4)^(3/4) = (2^3) = 8]
Now, let's calculate (3^4)/(2^3):
(3^4)/(2^3) = (81)/(8) = 10.125
Finally, let's simplify (8^3)^(2/3):
(8^3)^(2/3) = (2^3)^2 [since (8^3)^(2/3) = (2^3)^2]
(2^3)^2 = 2^(3*2) = 2^6 = 64
Now, let's multiply the results:
(16/81)^(-3/4) × 512^(2/3) = (10.125) × (64) = 648.