Emily claims that 323⋅33−−√6=323⋅336=3618=3–√3

Question

Is Emily correct? Justify your answer.

Please upload a picture of your handwritten work below.

Answer 1

To determine if Emily is correct, we need to evaluate the expression 323⋅33^(-6) and compare it with 3√3.

First, let's simplify the expression 323⋅33^(-6):

323⋅33^(-6) = 323⋅(3^3)^(-6) (since 33 can be written as 3^3)
= 323⋅3^(-18) (using the rule (a^b)^c = a^(b⋅c))
= 323/3^18 (using the rule a^(-b) = 1/(a^b))
= 323/3^(2⋅9) (since 18 can be written as 2⋅9)
= 323/((3^2)^9) (using the rule (a^b)^c = a^(b⋅c))
= 323/9^9 (simplifying the denominator)

Now, let's simplify 3√3:

3√3 = 3^(1/3) (since √3 can be written as 3^(1/2))
= (3^(1/2))^2/3 (using the rule √a = a^(1/2))
= (3^(2/2))/3 (simplifying the numerator and denominator separately)
= 3/3 (simplifying the exponent)
= 1

Since 323⋅33^(-6) simplifies to 323/9^9 and 3√3 simplifies to 1, we can see that Emily is incorrect. The two expressions are not equal.

Therefore, Emily's claim is incorrect.