What is (sqrt(18))^5 ?

The solution to (sqrt(18))^5 is 972sqrt(2)

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simplify (sqrt(18))^5

simplify

(18)^{5}

9722

Decimal Notation

1374.61558 \dots

Solution steps

(18)^{5}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

(18)^{4 + 1} = (18)^{4} 18

= (18)^{4} 18

(18)^{4} = ((18)^{2})^{2}

= ((18)^{2})^{2} 18

Apply exponent rule:

(18)^{2} = 18

= 1 8^{2} 18

Apply radical rule:

a = a^{\frac{1}{2}}

18 = 1 8^{\frac{1}{2}}

= 1 8^{2} \cdot 1 8^{\frac{1}{2}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

1 8^{2} \cdot 1 8^{\frac{1}{2}} = 1 8^{2 + \frac{1}{2}}

= 1 8^{2 + \frac{1}{2}}

2 + \frac{1}{2} = \frac{5}{2}

= 1 8^{\frac{5}{2}}

Prime factorization of

18 : 2 \cdot 3^{2}

= (2 \cdot 3^{2})^{\frac{5}{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(2 \cdot 3^{2})^{\frac{5}{2}} = 2^{\frac{5}{2}} (3^{2})^{\frac{5}{2}}

= 2^{\frac{5}{2}} (3^{2})^{\frac{5}{2}}

(3^{2})^{\frac{5}{2}} = 243

= 2^{\frac{5}{2}} \cdot 243

2^{\frac{5}{2}} = 2^{2} 2

= 2^{2} \cdot 2432

2^{2} = 4

= 4 \cdot 2432

Multiply the numbers:

4 \cdot 243 = 972

= 9722

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