Final answer:
To find the length x in simplest radical form with a rational denominator, algebraic methods such as factoring, using exponent rules, and setting up proportions are employed. Expressions are also rationalized by multiplying by a form of 1 to eliminate radicals in the denominator, and negative exponents are used to move terms between the numerator and the denominator.
Step-by-step explanation:
Finding the length x in simplest radical form with a rational denominator involves a series of algebraic steps, including factoring, using properties of exponents, and setting up ratios. For example, if we have the equation x² = √x, we know that x must be a perfect square in order for it to be its own square root, as implied by the fractional exponent. To simplify expressions with radicals and rationalize the denominator, we multiply by a form of 1 that will create a perfect square under the radical or eliminate the radical in the denominator entirely.
When dealing with ratios such as 0.5 inch/20 miles = 8 inches/x miles, we are setting up a proportion where we can cross-multiply to solve for the unknown x. This will give us the length in miles when we have a known scale length in inches. In the context of solving the equation (2x)² = 4.0 (1 − x)², we take the square root of both sides to simplify, and then rearrange the resulting equation to solve for x.
Moreover, understanding the concept of negative exponents, where x⁻¹ = 1/x, helps to simplify expressions and solve for unknown variables by moving terms between the numerator and the denominator. This knowledge is essential when multiplying and dividing units in ratio problems, making sure to cancel out units appropriately to leave the desired units for the answer.
By expanding expressions and multiplying both sides by the denominator, as shown in x² = 0.106 (0.360 - 1.202x+x²), we can solve for x in its simplest form while maintaining a rational denominator.