A. If b^2=a, then is a square root of . b. Given a>0, the...

a. If $b^{2}=a,$ then _______ is a square root of _______. b. Given $a>0,$ the symbol $\sqrt{a}$ denotes the positive or _______ square root of a. c. $b$ is an $n$ th root of $a$ if _______ = _______. d. Given the symbol $\sqrt[n]{a},$ the value $n$ is called the _______ and $a$ is called the _______. e. The symbol $\sqrt[3]{a}$ denotes the _______ root of $a$. f. The expression $\sqrt{-4}$ (is/is not) a real number. The expression $-\sqrt{4}$ (isfis not) a real number. g. The expression $\sqrt[n]{a^{n}}=|a|$ if $n$ is (even/odd). The expression $\sqrt[n]{a^{n}}=a$ if $n$ is (even/odd). h. Given a right triangle with legs $a$ and $b$ and hypotenuse $c,$ the _______ theorem is stated as $a^{2}+b^{2}=$ _______. i. In interval notation, the domain of $f(x)=\sqrt{x}$ is _______, whereas the domain of $g(x)=\sqrt[3]{x}$ is _______. j. Which of the following values of $x$ are not in the domain of $h(x)=\sqrt{x+3}: \quad x=-5, x=-4, x=-3, x=-2, x=-1, x=0 ?$

a. If $b^{2}=a,$ then _______ is a square root of _______.
b. Given $a>0,$ the symbol $\sqrt{a}$ denotes the positive or _______ square root of a.
c. $b$ is an $n$ th root of $a$ if _______ = _______.
d. Given the symbol $\sqrt[n]{a},$ the value $n$ is called the _______ and $a$ is called the _______.
e. The symbol $\sqrt[3]{a}$ denotes the _______ root of $a$.
f. The expression $\sqrt{-4}$ (is/is not) a real number. The expression $-\sqrt{4}$ (isfis not) a real number.
g. The expression $\sqrt[n]{a^{n}}=|a|$ if $n$ is (even/odd). The expression $\sqrt[n]{a^{n}}=a$ if $n$ is (even/odd).
h. Given a right triangle with legs $a$ and $b$ and hypotenuse $c,$ the _______ theorem is stated as $a^{2}+b^{2}=$ _______.
i. In interval notation, the domain of $f(x)=\sqrt{x}$ is _______, whereas the domain of $g(x)=\sqrt[3]{x}$ is _______.
j. Which of the following values of $x$ are not in the domain of $h(x)=\sqrt{x+3}: \quad x=-5, x=-4, x=-3, x=-2, x=-1, x=0 ?$

Key Concepts

Domain of Radical Functions

The domain of a function involving radicals depends on the type of root used. For square roots and other even-indexed roots, the radicand must be non-negative to yield a real number, while odd-indexed roots allow for a broader domain including all real numbers.

Pythagorean Theorem

The Pythagorean theorem establishes a fundamental relationship in right-angled triangles, stating that the sum of the squares of the two legs is equal to the square of the hypotenuse. This theorem is central to geometry and is a basis for many other results and applications in mathematics.

Absolute Value in Exponentiation with Even Root

When dealing with an even index, the operation of taking the nth root of a number raised to the nth power results in the absolute value of that number because both positive and negative values yield the same result when raised to an even power.

Nth Root

An nth root of a number is any number that, when raised to the power n, equals the given number. This concept generalizes square roots (where n equals 2) and is applicable for any positive integer n, though the principal value is typically chosen for even indices.

Principal Root

The principal root of a number is the non-negative value taken by convention when the equation b² = a has two solutions. It ensures consistency when defining functions like the square root function, which is important in calculations and graphs.

Square Root

A square root of a number is defined as any number which, when squared, gives the original number. In mathematics, when we refer to the square root, we often mean the principal (non-negative) square root, especially when dealing with real numbers.

Radical Notation

Radical notation uses symbols such as ? or ?[n] to denote roots of numbers. The radicand is the number under the radical, and when an index is specified (as in ?[n]), it indicates which power should equal the radicand, establishing a concise way to represent roots in mathematical expressions.

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