Help on radicals and log problems?

Question

i don't understand how to work out these problems:

Simplify:

1. (-3 x^-2)^-3

2.4th root of 32 + 5 4th root of 2

3. third root of 192

4. expand:
loga 8xy^2 / z^5

Answer 1

1. Simplify inside the parantheses first.

(-3 / x^-2) ^-3

Now distribute your exponent, remember your power rule.

-3^-3 / x^6

The negative exponents make the numerator go to the denominator and the exponent becomes positive.

1 / -3^3 * x^6

1 / -27x^6

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2.
= 4th root of 16 x 4th root of 2 + 5 4th root of 2
= 2 4th root of 2 + 5 4th root of 2
=7 4th root of 2

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3. 3rd root of 192.
=3rd root of (64 x 3)
= 3rd root of 64 x 3rd root of 3
= 4 3rd root of 3

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4. loga (8xy^2 / z^5)

Remember that division in the argument means you can subtract the logs..

loga (8xy^2) - loga (z^5)

Multiplication in the argument = add the logs...
Power of the whole argument = bring the power down in front.

loga 8 + loga x + loga (y^2) - 5loga z

loga 8 + loga x + 2loga y - 5loga z

Answer 2

Simplify:

1. (-3 x^-2)^-3 = (-3)^-3 x^{6} = -(x^6)/27

2.4th root of 32 + 5 4th root of 2

3. third root of 192 = 5.76

4. expand:
loga 8xy^2 / z^5 = log_a (8xy^2) - log_a (z^5)
= log_a (8) + log_a (x) + 2log_a (y) - 5log_a (z)
.

Answer 3

1.)
(-3x^-2)^(-3)
-3^(-3) * x^(-2 * -3)
(-1/(3^3)) * x^6
(-1/27) * x^6
(-x^6)/27

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2.)
4thrt(32) + 5(4thrt(2))
4thrt(16 * 2) + 5(4thrt(2))
2(4thrt(2)) + 5(4thrt(2))
7(4thrt(2))

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3.)
cbrt(192) = cbrt(64 * 3) = 4cbrt(3)

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4.)
log(a)(8xy^2)/(z^5))
log(a)(8xy^2) - log(a)(z^5)
log(a)8 + log(a)x + log(y^2) - 5log(a)z
log(a)8 + log(a)x + 2log(a)y - 5log(a)z