Question

What is a coefficient of a square root and why is it defaultly understood to be 1? Is it the same as the index or the exponent? Also how is an exponent with a coefficient read?

Answer 1

The coefficient of any designated quantity is the multiplier it has in the term of interest. If the quantity is the term of interest, its multiplier (coefficient) is 1.

Explanation:

The multiplicative identity element is 1. Anything multiplied by 1 is that thing. So, anything can be considered to have a multiplier of 1 at any time for any reason.

When an expression consists of a variable or some product of variables, for example, x or xz³, the multiplier of 1 is considered to be implicit. (The existence of the term means there is one of it.) The multiplier of 1 is usually not shown. If the multiplier is some other number, then, of course, it must be shown.

For expressions such as "mx + b", the term mx can be considered different ways, depending on the need. One could say the coefficient of the term mx is 1. The coefficient of x is m. The coeffcient of m is x. Note that in each case, we have designated something as having a coefficient, and the coefficient is everything else that multiplies that something.

If your something is a square root, the notions still apply. In the term √8, the coefficient of √12 is 1. If it were rewritten as 2√3, then the coefficient of the square root (√3) is 2. This multiplier is not the same as the index of the surd.

The index is the little number written in the notch of the surd symbol. In ∛, it is 3. In √, it is understood to be 2. This is the power to which the root must be raised to give the value under the surd symbol. Consequently, it is also the denominator of the fractional power represented by the root. √5 = 5^(1/2). ∛5 = 5^(1/3), for example.

`\left(\sqrt[3]{5}\right)^(3)=\left(5^{(1)/(3)}\right)^(3)=5^{(3)/(3)}=5`