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Examples
(a)
root(5)(-32)
(b)
-root(5)(32)
(c)
root(4)(81)
(d)
sqrt81
(e)
sqrt(-32)

Examples $\newline$ (a) $\sqrt[5]{-32}$ $\newline$ (b) $-\sqrt[5]{32}$ $\newline$ (c) $\sqrt[4]{81}$ $\newline$ (d) $\sqrt{81}$ $\newline$ (e) $\sqrt{-32}$

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Simplify the product of two radical expressions having same variable

Full solution

Q. Examples

\newline

(a)

\sqrt[5]{-32}

\newline

(b)

-\sqrt[5]{32}

\newline

(c)

\sqrt[4]{81}

\newline

(d)

\sqrt{81}

\newline

(e)

\sqrt{-32}

Simplify $-32$ : (a) Simplify $\sqrt[5]{-32}$ . $\newline$ Calculation: $\sqrt[5]{-32} = (-32)^{1/5}$ . $\newline$ Since the index is odd, the root of a negative number is defined. $\newline$ $(-32)^{1/5} = -2$ because $(-2)^5 = -32$ .
Simplify $-32$ : (b) Simplify $-\sqrt[5]{32}$ . $\newline$ Calculation: $\sqrt[5]{32} = 32^{1/5}$ . $\newline$ $32^{1/5} = 2$ because $2^5 = 32$ . $\newline$ Then, $-\sqrt[5]{32} = -2$ .
Simplify $81$ : (c) Simplify $\sqrt[4]{81}$ . $\newline$ Calculation: $\sqrt[4]{81} = 81^{\frac{1}{4}}$ . $\newline$ $81^{\frac{1}{4}} = 3$ because $3^4 = 81$ .
Simplify $81$ : (d) Simplify $\sqrt{81}$ . $\newline$ Calculation: $\sqrt{81} = 81^{1/2}$ . $\newline$ $81^{1/2} = 9$ because $9^2 = 81$ .
Simplify $-32$ : (e) Simplify $\sqrt{-32}$ . $\newline$ Calculation: $\sqrt{-32} = (-32)^{1/2}$ . $\newline$ Since the index is even, the square root of a negative number is not defined in the real number system.

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