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.
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To determine whether Emily's claim is correct, let's first simplify both sides of the equation.
Starting with the left side:
3^(2/3) * root(6, 3^3)
We can rewrite 3^3 as (3^2) * 3, since (a^b)^c = a^(b * c):
3^(2/3) * root(6, (3^2) * 3)
Next, we can apply the properties of exponents to simplify further. By multiplying exponents with the same base, we can combine them:
(3^2/3) * root(6, 3^2) * root(6, 3)
Using the property that the nth root of a product is equal to the product of the nth roots, we get:
(3^2/3) * root(6, 3^2 * 3)
(3^2/3) * root(6, 9 * 3)
(3^2/3) * root(6, 27)
Now, let's simplify the right side of the equation:
3^(2/3) * 3^3/6
Using the property that (a^b)/(a^c) = a^(b-c), we can simplify the exponent of 3:
3^(2/3) * 3^1/6
Notice that 3^1/6 can also be rewritten as root(6, 3).
Comparing the simplified expressions from both sides, we have:
(3^2/3) * root(6, 27) = 3^(2/3) * root(6, 3)
Therefore, Emily's claim is correct as the left side expression is equal to the right side expression.
Justify your answer.
To justify our answer, we can show that the left side expression is equal to the right side expression by simplifying both sides separately and then comparing them.
Starting with the left side expression:
We have 3^(2/3) * root(6, 3^3)
Using the properties of exponents, we can rewrite 3^3 as (3^2) * 3
Then, we apply the property that the nth root of a product is equal to the product of the nth roots, which gives us:
(3^(2/3)) * root(6, (3^2) * 3)
Simplifying further:
(3^(2/3)) * root(6, 9 * 3)
(3^(2/3)) * root(6, 27)
Now, let's move on to the right side expression:
We have 3^(2/3) * 3^(3/6)
Using the property that (a^b)/(a^c) = a^(b-c), we can simplify the exponent of 3:
3^(2/3) * 3^(1/6)
Notice that 3^(1/6) can also be written as root(6, 3).
Comparing the simplified expressions from both sides, we have:
(3^(2/3)) * root(6, 27) = 3^(2/3) * root(6, 3)
Therefore, we can conclude that Emily is correct because the left side expression is indeed equal to the right side expression through simplification and rearrangement of terms.
Justify your answer and Simplify
Apologies for not providing a simplified version of the expressions earlier. Let's continue the justification and simplify the expressions.
Starting with the left side expression:
(3^(2/3)) * root(6, 27)
We can rewrite 27 as 3^3:
(3^(2/3)) * root(6, 3^3)
Using the property that the nth root of a power is equal to the power divided by n, we have:
(3^(2/3)) * (3^3)^(1/6)
Applying the power of a power property, which states that (a^b)^c is equal to a^(b*c), we get:
(3^(2/3)) * 3^(3/6)
Simplifying further, we can evaluate the powers of 3:
(3^(2/3)) * 3^(1/2)
To multiply the two exponential terms with the same base, we can add the exponents:
3^(2/3 + 1/2)
Finding a common denominator for 3/2 and 3/6 gives us:
3^(4/6 + 3/6)
Adding the fractions:
3^(7/6)
Now, let's simplify the right side expression:
3^(2/3) * 3^(3/6)
Following the same steps as before, we can add the exponents:
3^(2/3 + 3/6)
Finding a common denominator for 3/2 and 3/6 gives us:
3^(4/6 + 3/6)
Adding the fractions:
3^(7/6)
After we simplify both sides, we see that the left side expression and the right side expression are equal to 3^(7/6).
Therefore, Emily is correct, and the simplified expression is 3^(7/6).