Question

(a) What does$\sqrt[n]{a}=b$ mean?

(b) Is it true that$\sqrt{a^2}$ is equal to $\left|a\right|$? Try values for a that are positive and negative.

(c) How many real$n^{th}$ root does a positive real number if n is even? If n is odd?

(c) What does$a^{-n}$ mean if n is negative integer? What does$3^{-2}$ mean?

(d) Is$\sqrt[4]{-2}$ a real number? Is$\sqrt[3]{-2}$ a real number? Explain why or why not.

Short answer

(a) The expression $\sqrt[n]{a}=b$ means, b is the $n^{th}$ root of a.

(b)The given expression$\sqrt{a^2}=\left|a\right|$

(c)Thus, there two real$n^{th}$ roots if n is even and only one real$n^{th}$ root if n is odd.

(d)The expression$\sqrt[4]{-2}$1 is not defined for real number but the expression$\sqrt[3]{-2}$ is defined for the real number.

Step-by-step solution

Part a. Step 1. Define exponent.

The exponent of a number means how many time to use that number in a multiplication.

It is written as a small number to the right and above the base number.

Part a. Step 2. Define base.

The base is defines as the number that gets multiplied in an exponent.

Part a. Step 3. Find the mean of the given expression.

The given number is,

$\sqrt[n]{a}=b$

Here, n is the positive integer that is called the exponent and a,b are real number that is base.

So,$\sqrt[n]{a}=b$ is defined as b is the$n^{th}$ root of a.

$\sqrt[1]{a}\times \sqrt[2]{a}\times \sqrt[3]{a}\mathrm{...}\sqrt[n]{a}=b$

Thus, the expression$\sqrt[n]{a}=b$ means, b is the$n^{th}$ root of a.

Part b. Step 1. Define exponent and base.

The exponent of a number means how many time to use that number in a multiplication.

It is written as a small number to the right and above the base number.

The base is defines as the number that gets multiplied in an exponent.

Part b. Step 2. Define meaning of absolute.

For a real number a its absolute value is defined as,

$\left|a\right|=\left\{\begin{array}{l}a,a\ge 0\\ -a,a<0\end{array}\right.$

This means$\left|a\right|$ always gives value greater than or equal to zero.

Part b. Step 3. Define the meaning of given term.

The given number is,

$\sqrt{a^2}$

This means$\sqrt{a^2}$gives the positive value, because the square of any number gives the positive values.

Thus, the given expression width="73">a2=a

Part c. Step 1. Define exponent and base.

The exponent of a number means how many times to use that number in a multiplication.

It is written as a small number to the right and above the base number.

The base is defines as the number that gets multiplied in an exponent.

Part c. Step 2. When n even.

If n is even then every positive real number has two square roots, one is positive and another is negative.

Example:

The square root of 81 is 9 and $-9$, which have two square root one is positive and other is negative.

Part c. Step 3. When n is odd.

If n is odd then every positive real number has only one positive root.

Example:

The cube root of 125 is only 5 and that is only one cube root.

Thus, there two real$n^{th}$ roots if n is even and only one real$n^{th}$ root if n is odd.

Part d. Step 1. Define exponent and base.

The exponent of a number means how many time to use that number in a multiplication.

It is written as a small number to the right and above the base number.

The base is defines as the number that gets multiplied in an exponent.

Part d. Step 2. Find whether the expression is real or not.

Consider the expression.

$\sqrt[4]{-2}$

The above expression is not real number, because the fourth root of the negative number is that the four negative always cancel to give plus and it is not defined.

Part d. Step 3. Find whether the expression is real or not.

The given number is,

$\sqrt[3]{-2}$

The above expression is a real number because the third root of the negative number is that three negatives, gives a minus and it is defined.

Thus, the expression$\sqrt[4]{-2}$1 is not defined for real number but the expression$\sqrt[3]{-2}$ is defined for the real number.