Negative and Rational (Fractional) Exponents worksheet | Live ... - Free Printable

To solve the given problems involving negative and rational exponents, we need to carefully apply the rules of exponents. Let's go through each problem step by step.

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1. Simplify \((-9)^2\)

- The expression \((-9)^2\) means \((-9) \times (-9)\).
- When you multiply two negative numbers, the result is positive.
- Therefore, \((-9)^2 = 81\).

Correct Answer: \(3. 81\)

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2. Simplify \(-9^2\)

- The expression \(-9^2\) means the negative of \(9^2\). Here, the exponent applies only to \(9\), not to \(-9\).
- First, calculate \(9^2 = 9 \times 9 = 81\).
- Then, take the negative: \(-9^2 = -81\).

Correct Answer: \(4. -81\)

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3. Simplify \(-9^{-2}\)

- The expression \(-9^{-2}\) means the negative of \(9^{-2}\). Here, the exponent \(-2\) applies only to \(9\), not to \(-9\).
- Recall that \(a^{-n} = \frac{1}{a^n}\). So, \(9^{-2} = \frac{1}{9^2}\).
- Calculate \(9^2 = 9 \times 9 = 81\).
- Therefore, \(9^{-2} = \frac{1}{81}\).
- Taking the negative: \(-9^{-2} = -\frac{1}{81}\).

Correct Answer: \(2. -\frac{1}{81}\)

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4. Simplify \(-9^{\frac{1}{2}}\)

- The expression \(-9^{\frac{1}{2}}\) means the negative of \(9^{\frac{1}{2}}\). Here, the exponent \(\frac{1}{2}\) applies only to \(9\), not to \(-9\).
- Recall that \(a^{\frac{1}{2}} = \sqrt{a}\). So, \(9^{\frac{1}{2}} = \sqrt{9}\).
- Calculate \(\sqrt{9} = 3\).
- Taking the negative: \(-9^{\frac{1}{2}} = -3\).

Correct Answer: \(6. -3\)

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5. Simplify \(-9^{-\frac{1}{2}}\)

- The expression \(-9^{-\frac{1}{2}}\) means the negative of \(9^{-\frac{1}{2}}\). Here, the exponent \(-\frac{1}{2}\) applies only to \(9\), not to \(-9\).
- Recall that \(a^{-\frac{1}{2}} = \frac{1}{a^{\frac{1}{2}}}\). So, \(9^{-\frac{1}{2}} = \frac{1}{9^{\frac{1}{2}}}\).
- Calculate \(9^{\frac{1}{2}} = \sqrt{9} = 3\).
- Therefore, \(9^{-\frac{1}{2}} = \frac{1}{3}\).
- Taking the negative: \(-9^{-\frac{1}{2}} = -\frac{1}{3}\).

Correct Answer: \(8. -\frac{1}{3}\)

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6. Simplify \(\frac{1}{(-9)^2}\)

- The expression \(\frac{1}{(-9)^2}\) means the reciprocal of \((-9)^2\).
- First, calculate \((-9)^2 = (-9) \times (-9) = 81\).
- Then, take the reciprocal: \(\frac{1}{(-9)^2} = \frac{1}{81}\).

Correct Answer: \(1. \frac{1}{81}\)

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Final Answers:

1. \((-9)^2 = 81\) → 3. 81
2. \(-9^2 = -81\) → 4. -81
3. \(-9^{-2} = -\frac{1}{81}\) → 2. -\(\frac{1}{81}\)
4. \(-9^{\frac{1}{2}} = -3\) → 6. -3
5. \(-9^{-\frac{1}{2}} = -\frac{1}{3}\) → 8. -\(\frac{1}{3}\)
6. \(\frac{1}{(-9)^2} = \frac{1}{81}\) → 1. \(\frac{1}{81}\)

Boxed Final Answer:
\[
\boxed{3, 4, 2, 6, 8, 1}
\]