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Find the value of $x$ so that $3^x = (-27)^{-6}$

User Meredith
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Final answer:

To find the value of $x$ so that 3^x = (-27)^{-6}, we recognize that -27 is (-3)^3. Raising a power to a power multiplies the exponents, leading to 3^x = 3^{-18}. Thus, $x$ equals -18.

Step-by-step explanation:

The question asks us to find the value of $x$ that satisfies the equation 3^x = (-27)^{-6}.

To solve this, we first recognize that -27 is the same as -3 cubed, or (-3)^3. When we raise a power to another power, we multiply the exponents. So (-27)^{-6} becomes [(-3)^3]^{-6}, which simplifies to (-3)^{-18}. Because the base -3 is raised to a negative exponent, the expression is equivalent to 1 divided by (-3)^{18}, and the negative sign disappears since any number raised to an even power is positive.

Now, (-3)^{18} can be written as (3^1)^{18} which simplifies to 3^{18}. At this point, we have 3^x = 1/(3^{18}), which implies that 3^x = 3^{-18}. Therefore, the value of $x$ is -18.

User WhiteShadow
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