Question

Sample Response: Squaring and square root are inverses, so one should undo the other. That is, squaring the square root of a number results in the number. Using the power of a power rule, you multiply the exponents. Since a number to the first power is itself, the product of the exponents must equal 1. This means that the power of the square root must be the reciprocal of 2, or one half.

What did you include in your response? Check all that apply.
a) Use the power of a power rule to multiply the exponents.
b) The product of the exponents must be 1.
c) The exponents are reciprocals.

Answer 1

The processes of squaring and taking the square root are inverse operations. When taking the square root of an exponential, halve the exponent and extract the square root of the digit term. Inversion also applies to exponents, with negative powers representing reciprocals.

Step-by-step explanation:

To find the side length a of a right triangle, we usually start with the Pythagorean Theorem which states that a² + b² = c², where b is the other side and c is the hypotenuse. If we need to find a, and we have a², we must take the square root of a² to undo the squaring. In terms of exponents, squaring is raising to the power of 2, and taking the square root is equivalent to raising to the power of one-half, since (x²)^(1/2) = x^(2*1/2) = x^(1) = x. The processes of squaring and taking the square root are inverse operations.

Squaring of Exponentials consists of squaring the digit term normally and multiplying the exponent by 2. Conversely, when taking Square Roots of Exponentials, you would extract the square root of the digit term and halve the exponent, ensuring it's evenly divisible by 2 beforehand if necessary.

Understanding the inversion of exponents is also crucial. For example, x to the negative power is the same as 1 divided by x to the positive power. This can be applied in the context where you need to get the reciprocal of an exponential, such as finding the inverse of a square by taking the square root.