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Popular >

simplify sqrt(8)^7

Solution

simplify

8^{7}

Solution

10242

+1

Decimal Notation

1448.15468 \dots

Solution steps

(8)^{7}

Apply exponent rule:

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

(8)^{6 + 1} = (8)^{6} 8

= (8)^{6} 8

(8)^{6} = ((8)^{2})^{3}

= ((8)^{2})^{3} 8

Apply exponent rule:

(8)^{2} = 8

= 8^{3} 8

Apply radical rule:

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

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

= 8^{3} \cdot 8^{\frac{1}{2}}

Apply exponent rule:

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

8^{3} \cdot 8^{\frac{1}{2}} = 8^{3 + \frac{1}{2}}

= 8^{3 + \frac{1}{2}}

3 + \frac{1}{2} = \frac{7}{2}

= 8^{\frac{7}{2}}

Prime factorization of

8 : 2^{3}

= (2^{3})^{\frac{7}{2}}

Apply exponent rule:

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

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

= (2^{2} \cdot 2)^{\frac{7}{2}}

Apply exponent rule:

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

(2^{2} \cdot 2)^{\frac{7}{2}} = (2^{2})^{\frac{7}{2}} \cdot 2^{\frac{7}{2}}

= (2^{2})^{\frac{7}{2}} \cdot 2^{\frac{7}{2}}

(2^{2})^{\frac{7}{2}} = 128

= 128 \cdot 2^{\frac{7}{2}}

2^{\frac{7}{2}} = 2^{3} 2

= 2^{3} \cdot 1282

Factor integer

128 = 2^{7}

= 2^{7} \cdot 2^{3} 2

Apply exponent rule:

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

2^{7} \cdot 2^{3} = 2^{7 + 3}

= 2^{7 + 3} 2

Add the numbers:

7 + 3 = 10

= 2^{10} 2

2^{10} = 1024

= 10242

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Frequently Asked Questions (FAQ)

What is sqrt(8)^7 ?
The solution to sqrt(8)^7 is 1024sqrt(2)