Question

Suppose that a friend missed class and never learned what 37^1/3 means. Use exponent rules your friend would already know to calculate (37^1/3)^3. Explain why this means that 37^1/3 is the cube root of 37

Answer 1

To calculate (37^1/3)^3, the exponents are multiplied, resulting in 37^1, which equals 37. This reveals that 37^1/3 is the cube root of 37, as cubing the cube root returns the original number.

Step-by-step explanation:

To calculate (37^1/3)^3, we use the exponent rules that your friend already knows. An exponent raised to another exponent means you multiply the exponents. In this case, (1/3)*3 equals 1, simplifying our expression to 37^1, which is just 37. This demonstrates that 37^1/3 is indeed the cube root of 37 because cubing it returns the original number, 37.

Understanding the exponentiation process, let's break it down: (37^1/3)^3 is the same as 37^(1/3*3), and when you multiply 1/3 by 3, you get 1. So, the expression simplifies to 37^1, which is just 37, and therefore, 37^1/3 must equal the cube root of 37 because when you cube the cube root of a number, you get the original number back. The process of exponentiation and cubing of exponentials demonstrates how exponents and roots are inversely related operations.

Answer 2

the calculation
`\( (37^(1/3))^(3) \)` shows that
`\( 37^(1/3) \)` must be the cube root of 37 because it is the number that, when raised to the third power, equals 37. This is why
`\( 37^(1/3) \)` is the same as the cube root of 37.

To explain this concept to your friend who missed class, let's start with the basic exponent rules they would know. One of the fundamental rules of exponents is that when you raise a power to another power, you multiply the exponents. So if we have
`\( (x^(a))^(b) \)`, the result is
`\( x^(a \cdot b) \)`.

Now, let's apply this rule to the expression
`\( (37^(1/3))^(3) \)`:

1. We multiply the exponents
`\( (1)/(3) \)` and
`\( 3 \)` together:

`\[ (37^(1/3))^3 = 37^((1/3 \cdot 3)) = 37^1 \]`

2.
`\( 37^1 \)` simply means 37 to the power of 1, which is just 37 itself:

`\[ 37^(1) = 37 \]`

This demonstrates that
`\( (37^(1/3))^(3) \)` results in the original number, 37. This is because the exponent
`\( 1/3 \)` is the reciprocal of 3. When we raise 37 to the
`\( 1/3 \)` power, we are essentially finding a number that, when multiplied by itself three times (which is what the cube of a number means), gives us 37 back. Therefore,
`\( 37^(1/3) \)` is indeed the cube root of 37, because cubing the cube root of a number returns the original number.

So, the calculation
`\( (37^(1/3))^(3) \)` shows that
`\( 37^(1/3) \)` must be the cube root of 37 because it is the number that, when raised to the third power, equals 37. This is why
`\( 37^(1/3) \)` is the same as the cube root of 37.