How do you simplify (8/27)^(-2/3)(827)23?

2 Answers
Feb 14, 2017

9/494

Explanation:

(8/27)^(-2/3)(827)23
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=(27/8)^(2/3)=(278)23
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The prime factorization of 27 and 8 is:
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27=9xx3=3xx3xx3=3^327=9×3=3×3×3=33
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8=4xx2=2xx2xx2=2^38=4×2=2×2×2=23

Substituting the factorization on the above fraction we have:
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(27/8)^(2/3)(278)23
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=(3^3/2^3)^(2/3)=(3323)23
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=((3/2)^3)^(2/3)=((32)3)23
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Applying the property of power of a power that says:
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color(red)((a^n)^m=a^(mxxn))(an)m=am×n
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((3/2)^3)^(2/3)((32)3)23
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=(3/2)^(3xx(2/3))=(32)3×(23)
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=(3/2)^2=(32)2
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=9/4=94

Feb 14, 2017

=9/4=94

Explanation:

There are 3 different processes indicated in this expression.

Laws of indices:

x^-m = 1/x^m" "and" "(a/b)^-m = (b/a)^mxm=1xm and (ab)m=(ba)m

The second law is the one we will apply.

Also x^(p/q) = rootq(x)^pxpq=qxp

(8/27)^(-2/3) = (27/8)^(+2/3)(827)23=(278)+23

= root3(27/8)^2" "larr=32782 find the cube roots first

=(3/2)^2=(32)2

=9/4=94