Final answer:
The statement 4√2 equals 21/4 illustrates the relationship between roots and fractional exponents, showing that the fourth root of a number can be expressed as the number raised to the 1/4 power.
Step-by-step explanation:
The statement 4√2 = 21/4 can be simplified and better understood by comparing the relationship between exponents and roots. Let's use a similar example, where 51 · 51 = 51+1 = 52, which equals the square root of 5 when multiplied by itself.
This equivalence between exponents and roots can be represented as x2 having the same value as √x. Following this logic, when dealing with the fourth root, such as 4√2, it can be expressed as an exponent of 1/4, which gives us 21/4. To expand our understanding, raising a number to an integer power, like 43, is simply shorthand for repeatedly multiplying that number by itself. Thus, integer powers can also relate back to roots and fractional powers.
Let's clarify the relationship between a scale factor and the proportional sizes in a scale model. If the scale factor is 1:4 and the scale dimension is 4, the actual dimension would be found using the proportion 1:4 = 4:x, with x being the actual dimension. This method illustrates how we use mathematical concepts like proportions and exponents to solve real-world problems.