In Exercises $$ 7-21 $$, simplify the expression. Write your answer using only positi exponents. $$ \begin{array}{lll} ext { 7. } \frac{7^{-2} m^{0}}{n^{-4}} & ext { 8. } \frac{(-9)^{0} j^{-1} k^{-4}}{2^{0}} & ext { 9. } \frac{5^{-2} w^{0}}{y^{-10}} \ \begin{array}{lll} ext { 10. } \frac{t^{-5}}{8^{-2} s^{-3}} & ext { 11. } \frac{3^{-2} a^{-1}}{9^{-1} b^{-2} c^{0}} & ext { 12. } \frac{17 x^{0} y^{-8}}{4^{-2} z^{-6}} \ ext { 13. }\left(p^{6} ight)^{3} & ext { 14. }\left(q^{-4} ight)^{5} & ext { 15. } 5^{3} \cdot 5^{-7} \ ext { 19. }\left(-8 t^{2} ight)^{3} & ext { 17. } \frac{x^{7}}{x^{4} \cdot x^{2}} & ext { 18. } \frac{v^{5} \cdot v^{3}}{v^{2}}\end{array}\end{array} \begin{array}{lll} ext { 16. }-4 \cdot(-4)^{-2} & ext { 21. }\left(\frac{1}{3 h^{5}} ight)^{-4}\end{array} $$

Question

In Exercises $$ 7-21 $$, simplify the expression. Write your answer using only positi exponents. $$ \begin{array}{lll}	ext { 7. } \frac{7^{-2} m^{0}}{n^{-4}} & 	ext { 8. } \frac{(-9)^{0} j^{-1} k^{-4}}{2^{0}} & 	ext { 9. } \frac{5^{-2} w^{0}}{y^{-10}} \ \begin{array}{lll}	ext { 10. } \frac{t^{-5}}{8^{-2} s^{-3}} & 	ext { 11. } \frac{3^{-2} a^{-1}}{9^{-1} b^{-2} c^{0}} & 	ext { 12. } \frac{17 x^{0} y^{-8}}{4^{-2} z^{-6}} \ 	ext { 13. }\left(p^{6}ight)^{3} & 	ext { 14. }\left(q^{-4}ight)^{5} & 	ext { 15. } 5^{3} \cdot 5^{-7} \ 	ext { 19. }\left(-8 t^{2}ight)^{3} & 	ext { 17. } \frac{x^{7}}{x^{4} \cdot x^{2}} & 	ext { 18. } \frac{v^{5} \cdot v^{3}}{v^{2}}\end{array}\end{array} \begin{array}{lll}	ext { 16. }-4 \cdot(-4)^{-2} & 	ext { 21. }\left(\frac{1}{3 h^{5}}ight)^{-4}\end{array} $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$x$$
Calculate or simplify the expression $$ (3 h^{5})^{4} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(3h^{5}\right)^{4}$$
- step1: Use the properties of exponents:
  $$3^{4}\left(h^{5}\right)^{4}$$
- step2: Evaluate the power:
  $$81h^{20}$$
Calculate or simplify the expression $$ (-8)^{3} * t^{6} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(-8\right)^{3}t^{6}$$
- step1: Simplify:
  $$-8^{3}t^{6}$$
- step2: Evaluate the power:
  $$-512t^{6}$$
Calculate or simplify the expression $$ q^{-20} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$q^{-20}$$
- step1: Simplify:
  $$\frac{1}{q^{20}}$$
Calculate or simplify the expression $$ p^{18} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$p^{18}$$
Calculate or simplify the expression $$ 3^{-2} * 9 * a^{1} * b^{2} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$3^{-2}\times 9a^{1}b^{2}$$
- step1: Evaluate:
  $$3^{-2}\times 9ab^{2}$$
- step2: Transform the expression:
  $$3^{-2}\times 3^{2}ab^{2}$$
- step3: Multiply the terms:
  $$3^{-2+2}ab^{2}$$
- step4: Add the numbers:
  $$3^{0}ab^{2}$$
- step5: Evaluate the power:
  $$ab^{2}$$
Calculate or simplify the expression $$ (-9)^{0} * 2^{0} * j^{-1} * k^{-4} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(-9\right)^{0}\times 2^{0}j^{-1}k^{-4}$$
- step1: Evaluate the power:
  $$1\times 2^{0}j^{-1}k^{-4}$$
- step2: Evaluate the power:
  $$1\times 1\times j^{-1}k^{-4}$$
- step3: Multiply the terms:
  $$j^{-1}k^{-4}$$
- step4: Simplify:
  $$\frac{1}{jk^{4}}$$
Calculate or simplify the expression $$ t^{-5} * 8^{2} * s^{3} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$t^{-5}\times 8^{2}s^{3}$$
- step1: Reorder the terms:
  $$8^{2}t^{-5}s^{3}$$
- step2: Evaluate the power:
  $$64t^{-5}s^{3}$$
- step3: Simplify:
  $$\frac{64s^{3}}{t^{5}}$$
Calculate or simplify the expression $$ 5^{-2} * y^{10} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$5^{-2}y^{10}$$
- step1: Evaluate the power:
  $$\frac{1}{25}y^{10}$$
Calculate or simplify the expression $$ 5^{3} * 5^{-7} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$5^{3}\times 5^{-7}$$
- step1: Multiply:
  $$5^{3-7}$$
- step2: Subtract the numbers:
  $$5^{-4}$$
- step3: Express with a positive exponent:
  $$\frac{1}{5^{4}}$$
- step4: Evaluate the power:
  $$\frac{1}{625}$$
Calculate or simplify the expression $$ v^{6} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$v^{6}$$
Calculate or simplify the expression $$ -4 * (-4)^{-2} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$-4\left(-4\right)^{-2}$$
- step1: Calculate:
  $$\left(-4\right)^{-1}$$
- step2: Determine the sign:
  $$-4^{-1}$$
- step3: Express with a positive exponent:
  $$\frac{-1}{4}$$
- step4: Rewrite the fraction:
  $$-\frac{1}{4}$$
Calculate or simplify the expression $$ 7^{-2} * n^{4} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$7^{-2}n^{4}$$
- step1: Evaluate the power:
  $$\frac{1}{49}n^{4}$$
Calculate or simplify the expression $$ 17 * 4^{2} * z^{6} * y^{8} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$17\times 4^{2}z^{6}y^{8}$$
- step1: Multiply the terms:
  $$272z^{6}y^{8}$$
Let's simplify each expression step by step, ensuring we only use positive exponents in the final answers.

1. **Expression:** $$ \frac{7^{-2} m^{0}}{n^{-4}} $$

- Since $$ m^0 = 1 $$, we can simplify to:
   $$
   \frac{7^{-2}}{n^{-4}} = 7^{-2} \cdot n^{4} = \frac{n^{4}}{7^{2}} = \frac{n^{4}}{49}
   $$

**Final Answer:** $$ \frac{n^{4}}{49} $$

2. **Expression:** $$ \frac{(-9)^{0} j^{-1} k^{-4}}{2^{0}} $$

- Since $$ (-9)^{0} = 1 $$ and $$ 2^{0} = 1 $$, we have:
   $$
   \frac{1 \cdot j^{-1} \cdot k^{-4}}{1} = j^{-1} \cdot k^{-4} = \frac{1}{jk^{4}}
   $$

**Final Answer:** $$ \frac{1}{jk^{4}} $$

3. **Expression:** $$ \frac{5^{-2} w^{0}}{y^{-10}} $$

- Since $$ w^0 = 1 $$, we simplify to:
   $$
   \frac{5^{-2}}{y^{-10}} = 5^{-2} \cdot y^{10} = \frac{y^{10}}{5^{2}} = \frac{y^{10}}{25}
   $$

**Final Answer:** $$ \frac{y^{10}}{25} $$

4. **Expression:** $$ \frac{t^{-5}}{8^{-2} s^{-3}} $$

- This simplifies to:
   $$
   t^{-5} \cdot 8^{2} \cdot s^{3} = \frac{8^{2} s^{3}}{t^{5}} = \frac{64 s^{3}}{t^{5}}
   $$

**Final Answer:** $$ \frac{64 s^{3}}{t^{5}} $$

5. **Expression:** $$ \frac{3^{-2} a^{-1}}{9^{-1} b^{-2} c^{0}} $$

- Since $$ c^0 = 1 $$, we have:
   $$
   \frac{3^{-2} a^{-1}}{9^{-1} b^{-2}} = 3^{-2} \cdot 9 \cdot a^{-1} \cdot b^{2} = 3^{-2} \cdot 3^{2} \cdot a^{-1} \cdot b^{2} = \frac{b^{2}}{a}
   $$

**Final Answer:** $$ ab^{2} $$

6. **Expression:** $$ \frac{17 x^{0} y^{-8}}{4^{-2} z^{-6}} $$

- Since $$ x^0 = 1 $$, we simplify to:
   $$
   \frac{17 \cdot y^{-8}}{4^{-2} \cdot z^{-6}} = 17 \cdot 16 \cdot y^{-8} \cdot z^{6} = 272 \cdot z^{6} \cdot y^{8}
   $$

**Final Answer:** $$ 272 z^{6} y^{8} $$

7. **Expression:** $$ (p^{6})^{3} $$

- Using the power of a power property:
   $$
   p^{6 \cdot 3} = p^{18}
   $$

**Final Answer:** $$ p^{18} $$

8. **Expression:** $$ (q^{-4})^{5} $$

- Again using the power of a power property:
   $$
   q^{-4 \cdot 5} = q^{-20} = \frac{1}{q^{20}}
   $$

**Final Answer:** $$ \frac{1}{q^{20}} $$

9. **Expression:** $$ 5^{3} \cdot 5^{-7} $$

- Using the property of exponents:
   $$
   5^{3 - 7} = 5^{-4} = \frac{1}{5^{4}} = \frac{1}{625}
   $$

**Final Answer:** $$ \frac{1}{625} $$

10. **Expression:** $$ (-8 t^{2})^{3} $$

- Expanding this gives:
    $$
    (-8)^{3} \cdot (t^{2})^{3} = -512 \cdot t^{6} = -512 t^{6}
    $$

**Final Answer:** $$ -512 t^{6} $$

11. **Expression:** $$ \frac{x^{7}}{x^{4} \cdot x^{2}} $$

- Simplifying gives:
    $$
    x^{7 - (4 + 2)} = x^{1} = x
    $$

**Final Answer:** $$ x $$

12. **Expression:** $$ \frac{v^{5} \cdot v^{3}}{v^{2}} $$

- This simplifies to:
    $$
    \frac{v^{5 + 3}}{v^{2}} = v^{8 - 2} = v^{6}
    $$

**Final Answer:** $$ v^{6} $$

13. **Expression:** $$ -4 \cdot (-4)^{-2} $$

- This simplifies to:
    $$
    -4 \cdot \frac{1}{(-4)^{2}} = -4 \cdot \frac{1}{16} = -\frac{4}{16} = -\frac{1}{4}
    $$

**Final Answer:** $$ -\frac{1}{4} $$

14. **Expression:** $$ \left(\frac{1}{3 h^{5}}\right)^{-4} $$

- This simplifies to:
    $$
    \left(3 h^{5}\right)^{4} = 3^{4} h^{20} = 81 h^{20}
    $$

**Final Answer:** $$ 81 h^{20} $$

These are the simplified expressions with only positive exponents.
Here are the simplified expressions with only positive exponents: 1. $$ \frac{n^{4}}{49} $$ 2. $$ \frac{1}{jk^{4}} $$ 3. $$ \frac{y^{10}}{25} $$ 4. $$ \frac{64 s^{3}}{t^{5}} $$ 5. $$ ab^{2} $$ 6. $$ 272 z^{6} y^{8} $$ 7. $$ p^{18} $$ 8. $$ \frac{1}{q^{20}} $$ 9. $$ \frac{1}{625} $$ 10. $$ -512 t^{6} $$ 11. $$ x $$ 12. $$ v^{6} $$ 13. $$ -\frac{1}{4} $$ 14. $$ 81 h^{20} $$

In Exercises , simplify the expression. Write your answer using only positi
exponents.

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In Exercises , simplify the expression. Write your answer using only positi exponents.