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Timothy evaluated the expression using \( x = 3 \) and \( y = -4 \):

1. \(\left(\frac{1}{3}\right) x^{-1} y^2\)
2. \(\left(\frac{1}{3}\right) 3^{-1}(-4)^2\)
3. \(\left(\frac{1}{3}\right)\left(\frac{1}{3^1}\right)(-4)^2\)
4. \(\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)(-16)\)
5. \(-\frac{16}{9}\)

Analyze Timothy's steps. Is he correct? If not, why not?

A. Yes, he is correct.
B. No, he needed to add the exponents when he simplified the powers of the same base.
C. No, he needed to multiply 3 and -1 instead of creating a positive exponent in a fraction.
D. No, his value of [tex]\((-4)^2\)[/tex] should be positive because an even exponent indicates a positive value.



Answer :

GinnyAnswer
Let's analyze Timothy's steps in detail to see if he correctly evaluated the expression given \( x = 3 \) and \( y = -4 \).

The expression to evaluate is: \(\left(\frac{1}{3}\right) x^{-1} y^2\).

Step-by-step solution:

1. Substitute \( x = 3 \) and \( y = -4 \) into the expression:
[tex]\[ \left(\frac{1}{3}\right) x^{-1} y^2 = \left(\frac{1}{3}\right) 3^{-1} (-4)^2 \][/tex]

2. Evaluate the powers:
- \( 3^{-1} = \frac{1}{3} \)
- \( (-4)^2 = 16 \)

So the expression becomes:
[tex]\[ \left(\frac{1}{3}\right) \left(\frac{1}{3}\right) \cdot 16 \][/tex]

3. Simplify the fractions and multiply:
[tex]\[ \left(\frac{1}{3}\right) \left(\frac{1}{3}\right) \cdot 16 = \frac{1}{3} \cdot \frac{1}{3} \cdot 16 = \frac{1}{9} \cdot 16 = \frac{16}{9} \][/tex]

4. Convert the fraction to a decimal:
[tex]\[ \frac{16}{9} \approx 1.7777777777777777 \][/tex]

So, the result of the evaluated expression is approximately \( 1.7777777777777777 \).

### Analysis of Timothy's Steps:

Let's consider each of Timothy's steps:

1. \(\left(\frac{1}{3}\right) x^{-1} y^2\)
2. \(\left(\frac{1}{3}\right) 3^{-1}(-4)^2\)
3. \(\left(\frac{1}{3}\right)\left(\frac{1}{3^1}\right)(-4)^2\)

These steps are correct so far.

4. \(\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)(-16)\)

Here, Timothy made a mistake. Evaluating \((-4)^2\) correctly should yield \(16\), not \(-16\).

Let's check the next suggestion in the question:
- No, his value of \( (-4)^2\) should be positive because an even exponent indicates a positive value.

This suggestion is correct. An even exponent makes the value positive, therefore:

Timothy's calculations were incorrect because the value of [tex]\( (-4)^2\)[/tex] should be positive 16.

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