Answer:
Explanation:
To simplify the expression [27/125] - (⅔) × [25/9] - (½) ÷ (27^(-⅓)), let's break it down step by step:
Step 1: Simplify the exponents
- The expression contains negative exponents, which need to be converted to positive exponents. To do this, we'll use the reciprocal property. The reciprocal of a number with a negative exponent is the same number with a positive exponent.
(27^(-⅓)) is equivalent to (1 / 27^(⅓)) because we flipped the fraction and made the exponent positive.
Now, the expression is:
[27/125] - (⅔) × [25/9] - (½) ÷ (1 / 27^(⅓))
Step 2: Evaluate the roots
- Calculate 27^(⅓), which is the cube root of 27.
27^(⅓) = 3, because 3 × 3 × 3 = 27.
Now, the expression is:
[27/125] - (⅔) × [25/9] - (½) ÷ (1 / 3)
Step 3: Calculate the fractions
- Simplify the fractions and mixed numbers:
[27/125] = (3 × 3 × 3) / (5 × 5 × 5) = 3^3 / 5^3
[25/9] = (5 × 5) / (3 × 3) = 5^2 / 3^2
(⅔) = 2/3
(½) = 1/2
The expression becomes:
(3^3 / 5^3) - (2/3) × (5^2 / 3^2) - (1/2) ÷ 3
Step 4: Perform multiplication and division
- Now, perform the multiplication and division:
= (3^3 / 5^3) - (2/3) × (5^2 / 3^2) - (1/2) ÷ 3
= (27/125) - (10/9) - (1/6)
Step 5: Find a common denominator
- To combine the fractions, find a common denominator, which is 18 in this case:
= (27/125) - (20/18) - (3/18)
Step 6: Combine the fractions
- Combine the fractions:
= (27/125) - (20/18) - (3/18)
Step 7: Simplify further
- Simplify the fractions:
= (27/125) - (10/9) - (1/2)
Now, the expression is fully simplified.
So, [27/125] - (⅔) × [25/9] - (½) ÷ (27^(-⅓)) simplifies to (27/125) - (10/9) - (1/2).