Chapter 6, Problem 1VR

Complete each statement with the correct term from the column on the right. Some of the choices may not be used and some may be used more than once.

radicand

index

square

complex

cube

conjugate

radical

principal

imaginary

rationalizing

odd

The number c is the ___________root of a, written $\sqrt[3]{a}$ if the third power of c is a—-that is, if $c^3=a,\text{ then }\sqrt[3]{a}=a$ then. [6.1c]

To determine

The correct term which completes the statement “The number c is the ____ root of a, written $a^3$, if the third power of c is a- that is, if $c^3=a$, then $\sqrt[3]{a}=c$ ” from the column,

Answer to Problem 1VR

Solution:

The correct term that completes the statement “The number c is the ____ root of a, written $a^3$, if the third power of c is a- that is, if $c^3=a$, then $\sqrt[3]{a}=c$ ” is cube, that is, option f).

Explanation of Solution

Given Information:

The provided options are,

a) i

b) radicand

c) index

d) square

e) complex

f) cube

g) conjugate

h) radical

i) principal

j) imaginary

k) rationalizing

l) odd

Consider the provided statement is:

The number c is the ____ root of a, written $a^3$, if the third power of c is a- that is, if $c^3=a$, then $\sqrt[3]{a}=c$.

Since,

For a real number a,

Consider another real number c,

If the third power of c is a- that is, if $c^3=a$, and $\sqrt[3]{a}=c$.

Then c is called the cube root of a.

Hence, the correct term that completes the statement is “cube”, which is same as option f).

Consider option a),

$i$

Since, the term $i$ is a complex number,

And does not complete the statement correctly.

Hence, option a) is incorrect.

Consider option b),

Radicand

Since, the radicand is the expression under the radical.

And does not complete the statement correctly.

Hence, option b) is incorrect.

Consider option c),

Index

Since, the index is the number provided to an expression.

And does not complete the statement correctly.

Hence, option c) is incorrect.

Consider option d),

Square

Since, the square root of a number exists when $c^2=a$, and $\sqrt{a}=c$,

And does not complete the statement correctly.

Hence, option d) is incorrect.

Consider option e),

Complex

Since, the term complex corresponds to complex numbers, Example $i$ etc.

And does not complete the statement correctly.

Hence, option e) is incorrect.

Consider option g),

Conjugate

Since, conjugate exists for complex numbers,

Ex- conjugate of $x+iy$ is $x-iy$.

And does not complete the statement correctly.

Hence, option g) is incorrect.

Consider option h),

Radical

Since, the radical is the expression $\sqrt{}$,

And does not complete the statement correctly.

Hence, option h) is incorrect.

Consider option i),

Principal

Since, principal is the root of the equation such that, non-negative numbers are included.

And does not complete the statement correctly.

Hence, option i) is incorrect.

Consider option j),

Imaginary

Since, imaginary roots are the roots which exist for equations which have no real roots.

And does not complete the statement correctly.

Hence, option j) is incorrect.

Consider option k),

Rationalizing

Since, rationalizing refers to removing irrational terms from the denominator of expressions.

And does not complete the statement correctly.

Hence, option k) is incorrect.

Consider option l),

Odd

Since, odd roots are of the type $\sqrt[3]{},\sqrt[5]{}$ and so on.

And does not complete the statement correctly.

Hence, option l) is incorrect.

The options a), b), c), d), e), g), h), i), j), k), l) do not match the criteria of the equation most easily solved by factorizing.

Hence options a), b), c), d), e), g), h), i), j), k), l) are not correct.

Therefore, the term cube, that is option f), is the correct term that completes the statement.