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Understanding the Expression (27*64)^1/3
The expression (27*64)^1/3 represents the cube root of the product of 27 and 64. Let's break down the meaning and solve it:
Cube Root
The exponent 1/3 indicates a cube root. A cube root of a number is a value that, when multiplied by itself three times, equals the original number.
For example, the cube root of 8 is 2 because 2 * 2 * 2 = 8.
Solving the Expression
- Calculate the product inside the parentheses: 27 * 64 = 1728
- Find the cube root of 1728: The cube root of 1728 is 12, since 12 * 12 * 12 = 1728.
Therefore, (27*64)^1/3 = 12
Simplifying the Expression
We can also simplify the expression by recognizing that 27 and 64 are perfect cubes:
- 27 is the cube of 3 (3 * 3 * 3 = 27)
- 64 is the cube of 4 (4 * 4 * 4 = 64)
Therefore, the expression can be rewritten as:
(3^3 * 4^3)^1/3
Using the properties of exponents, we can simplify further:
(3^3 * 4^3)^1/3 = 3^(31/3) * 4^(31/3) = 3^1 * 4^1 = 3 * 4 = 12
This demonstrates that we can solve the expression by identifying the perfect cubes within it and using the properties of exponents.