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Popular >

simplify sqrt(8)^5

Solution

simplify

8^{5}

Solution

1282

+1

Decimal Notation

181.01933 \dots

Solution steps

(8)^{5}

Apply exponent rule:

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

(8)^{4 + 1} = (8)^{4} 8

= (8)^{4} 8

(8)^{4} = ((8)^{2})^{2}

= ((8)^{2})^{2} 8

Apply exponent rule:

(8)^{2} = 8

= 8^{2} 8

Apply radical rule:

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

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

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

Apply exponent rule:

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

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

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

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

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

Prime factorization of

8 : 2^{3}

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

Apply exponent rule:

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

2^{3} = 2^{2} \cdot 2

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

Apply exponent rule:

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

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

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

(2^{2})^{\frac{5}{2}} = 32

= 32 \cdot 2^{\frac{5}{2}}

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

= 2^{2} \cdot 322

Factor integer

32 = 2^{5}

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

Apply exponent rule:

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

2^{5} \cdot 2^{2} = 2^{5 + 2}

= 2^{5 + 2} 2

Add the numbers:

5 + 2 = 7

= 2^{7} 2

2^{7} = 128

= 1282

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Frequently Asked Questions (FAQ)

What is sqrt(8)^5 ?
The solution to sqrt(8)^5 is 128sqrt(2)