Powers: Adding and Subtracting Powers
$${3}^{-6}+{3}^{-6}+{3}^{-6}=$$
Correct.
Begin by finding the common factor. Since all of the terms are identical, the common factor is $$3^{-6}$$ itself. Place the common factor to the left of a pair of brackets, and populate the brackets with remaining factors. In this case, the remaining factors are 1, since the common factor is identical to the terms (in other words, $$3^{-6}$$ needs to be multiplied by 1 to reach $$3^{-6}$$). >$$3^{-6} + 3^{-6} +3^{-6}$$ >$$3^{-6} \cdot (1+1+1)$$ >$$3^{-6} \cdot 3$$ >$$3^{-6} \cdot 3^1$$ Using the rule for multiplying powers of the same base, add the exponents. >$$3^{-6+1}=3^{-5}$$ Finally, switch the exponent from negative to positive by transforming the base into its reciprocal. >$$\displaystyle \frac{1}{3^5}$$ And that's it.Alternative Explanation: Extracting a common factor here is technically redundant. The sum of the term $$3^{-6}$$ three times is simply $$3^{-6}$$ multiplied by 3, just as $$x+x+x$$ is equivalent to $$3x$$.
Incorrect.
Most students who choose this trap answer do so because they've mistakenly added the exponents. Remember: to add powers of the same base, extract a common factor. Adding the exponents is done when the powers are multiplied (e.g., $$3^{-6} \cdot 3^{-6} \cdot 3^{-6}$$), not added.
Incorrect.
[[snippet]]Incorrect.
[[snippet]]Incorrect.
[[snippet]] While it is true that the question stem equals $$3(3^{-6})$$, most students who choose this trap answer choice do so because they mistakenly multiply the base of $$3\cdot3$$, ignoring the exponent. Note that this is the same as adding $$3+3+3$$. Are either of these what happened?A common, and dangerous, mistake. Note that you can't multiply a
power by another term by multiplying the base alone. $$3$$ times the power of $$3$$ is
not the same as $$3 \cdot 3$$. Actually, $$3^a$$ is $$3$$ multiplied by itself (i.e., $$3 \cdot 3 \cdot 3…$$) $$a$$ times. You cannot just multiply $$3$$ times a single $$3$$ and ignore the power.
Multiplying a power by a coefficient demands resolving the power first, or at least breaking it apart into something more manageable. The golden rule of powers is very simple: if it isn't defined as one of the power rules (i.e., multiplication = add the exponents, division =
subtract the exponent, and so on), don't do it.
Good job not falling into that trap. You must have made some other calculation error. Be careful.
$$\displaystyle \frac{1}{{9}^{6}}
$$
$$\displaystyle \frac{1}{{9}^{3}}$$
$$\displaystyle \frac{1}{{3}^{18}}$$
$$\displaystyle \frac{1}{{3}^{5}}$$
$$\displaystyle \frac{1}{{3}^{3}}$$
Oops…yes.
No.
