Emily claims that
3^2/3*root(6,3^3)=3^(2)/(3)*3^3/6=3(6)/(18)=root(3,3)
Is Emily correct? Justify your answer.
Please upload a picture of your handwritten work below.
(3 points)
To justify whether Emily's claim is correct, let us simplify each step of her work.
Step 1: 3^(2/3) * root(6, 3^3)
The cube root of 3^3 is 3, so this expression becomes:
3^(2/3) * 3
= 3^(2/3+1)
= 3^(5/3)
= (3^5)^(1/3)
= 243^(1/3)
= root(3, 243)
Step 2: 3^(2)/(3) * 3^3/6
Let's evaluate each part of this expression separately.
3^(2)/(3) = (3^2)/(3) = 9/3 = 3
3^3/6 = (3^3)/(6) = 27/6 = 9/2
Multiplying these two parts, we get:
3 * (9/2) = 27/2
As we see, the expression in the first step is not equal to the expression in the second step, so Emily's claim is NOT correct.
Justify your answer and Simplify
Certainly!
To justify our answer, let's simplify each step of Emily's work and compare the results.
Step 1: 3^(2/3) * ∛(3^3)
We can simplify 3^(2/3) as follows:
3^(2/3) = (3^(1/3))^2 = ∛(3)^2 = (√3)^2 = 3.
So, the expression becomes:
3 * ∛(3^3) = 3 * ∛(27) = 3 * 3 = 9.
Step 2: 3^(2)/(3) * 3^3/6
We can simplify this expression separately:
3^(2)/(3) = (3^2)/(3) = 9/3 = 3.
3^3/6 = (3^3)/(6) = 27/6 = 4.5.
Multiplying these two parts, we get:
3 * 4.5 = 13.5.
Comparing the results:
In Step 1, the expression is equal to 9.
In Step 2, the expression is equal to 13.5.
Since 9 is not equal to 13.5, Emily's claim is NOT correct.
Therefore, Emily's claim is incorrect.