Question

I need help on solving this kind of.

Answer 1

81

Explanation:

The relevant rules of exponents are ...

(ab)^c = (a^c)(b^c)

a^-b = 1/a^b

(a^b)(a^c) = a^(b+c)

(a^b)^c = a^(bc)

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Applying these rules, we can simplify this expression as follows:

`((3x^(-4))^6)/(9x^(-12))=(3^6)/(9)x^((-4)(6)-(-12))=81x^0=\boxed{81}`

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Additional comment

The rules of exponents derive from the fact that an exponent indicates repeated multiplication. That is, x·x·x = x³, where the exponent 3 indicates x is a factor 3 times. When this idea is used for division, we see the exponent is the difference between numerator and denominator exponents:

(x·x·x)/(x·x) = x or x³/x² = x³⁻² = x¹ = x

This idea extends to cases where the denominator exponent is larger:

(x·x)/(x·x·x) = 1/x or x²/x³ = x²⁻³ = x⁻¹ = 1/x

This can help you see that moving a factor from numerator to denominator (or vice versa) changes the sign of its exponent.

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This idea of repeated multiplication also applies when exponents are layered:

(x·x·x)(x·x·x) = (x³)² = x⁶ . . . . . the product exponent is the product of 3 and 2 as we indicated in the rules above