- Exponents and roots are not distributive oras $$ (a+b)^{n}
eq a^{n}+b^{n}, \quad(a-b)^{n}
eq a^{n}-b^{n}, \quad \sqrt[n]{a+b}
eq \sqrt[n]{a}+\sqrt[n]{b} $$ and $$ \sqrt[n]{a-b}
eq \sqrt[n]{a} $$. EXERCISE 2 (a) Simplify the following: (2) $$ 2 x^{3}+2 x^{3} $$ (1) $$ 2 x^{3} imes 2 x^{3} $$ (2) $$ 2 x^{3}+2 x^{3} $$ (4) $$ 3 x^{3} imes 2 x^{3}+6 x^{2} imes 2 x^{4} $$ (3) $$ \left(3 x^{3} imes 2 x^{3} ight)+\left(3 x^{3}+2 x^{3} ight) $$ (b) Simplify the following, leaving your answer with positive exponents: ((3) $$ \frac{a^{6} b^{-8} c}{a^{-8} b^{6} c^{-3}} $$ (1) $$ \frac{x^{-12}}{x^{-7}} $$ (2) $$ \frac{x^{-7}}{x^{-12}} $$ (c) Simplify the following, leaving your answers with positive exponents: (1) $$ \left(x^{-3} ight)^{4} $$ (2) $$ \left(x^{2} ight)^{-5} $$ (3) $$ \left(x^{-2} ight)^{-2} $$ (4) $$ \left(-4 x^{-3} ight)^{3} $$ (5) $$ \left(-2 x^{3} ight)^{4} $$ (6) $$ \left(x^{4} ight)^{2}+\left(x^{2} ight)^{4} $$ (7) $$ 2\left(x^{4} ight)^{2}+4\left(x^{2} ight)^{4} $$ (8) $$ \left(-x^{3} ight)^{2} \cdot\left(-x^{2} ight)^{3} $$ (9) $$ 2\left(-x^{4} ight)^{2} imes 4\left(-x^{2} ight. $$ (10) $$ \left(3 x^{4} ight)^{2}-\left(2 x^{2} ight)^{4} $$ (11) $$ 3\left(-2 a^{3} ight)^{2}-2\left(-3 a^{2} ight)^{3} $$ (12) $$ \left(3 x^{4} ight)^{2} \cdot\left(2 x^{2} ight)^{4} $$ (13) $$ 3\left(-2 a^{3} ight)^{2} imes 2\left(-3 a^{2} ight)^{3} $$ (d) Simplify the following, leaving your answers with positive exponents: (1) $$ \left(\frac{a^{4}}{b^{6}} ight)^{2} $$ (2) $$ \left(\frac{10 x^{3} y}{5 x y^{2}} ight)^{3} $$ (3) $$ \left(\frac{-2 p^{4} q^{-2}}{4 p^{-6}} ight)^{3} $$ (4) $$ \left(\frac{3 x^{-5}}{9 x^{7}} ight)^{-2} $$ (5) $$ \left(\frac{2 a^{3} \cdot 4 a^{2}}{8\left(a^{-3} ight)^{2}} ight)^{2} $$ (6) $$ \left(\frac{x^{2}+x^{2}}{x^{2} imes x^{2}} ight)^{2} $$ $$ \frac{\left(x^{-3} y^{5} ight)^{2}}{x^{3} y^{-6}} $$ (81) $$ \frac{-3\left(a^{-1} b^{2} ight)^{-3} imes(a b)^{-5}}{\left(-3 b^{-5} ight)^{2}} $$ (9) $$ \frac{x+y}{x^{-1}+y^{-1}} $$

Question

- Exponents and roots are not distributive oras $$ (a+b)^{n} 
eq a^{n}+b^{n}, \quad(a-b)^{n} 
eq a^{n}-b^{n}, \quad \sqrt[n]{a+b} 
eq \sqrt[n]{a}+\sqrt[n]{b} $$ and $$ \sqrt[n]{a-b} 
eq \sqrt[n]{a} $$. EXERCISE 2 (a) Simplify the following: (2) $$ 2 x^{3}+2 x^{3} $$ (1) $$ 2 x^{3} 	imes 2 x^{3} $$ (2) $$ 2 x^{3}+2 x^{3} $$ (4) $$ 3 x^{3} 	imes 2 x^{3}+6 x^{2} 	imes 2 x^{4} $$ (3) $$ \left(3 x^{3} 	imes 2 x^{3}ight)+\left(3 x^{3}+2 x^{3}ight) $$ (b) Simplify the following, leaving your answer with positive exponents: ((3) $$ \frac{a^{6} b^{-8} c}{a^{-8} b^{6} c^{-3}} $$ (1) $$ \frac{x^{-12}}{x^{-7}} $$ (2) $$ \frac{x^{-7}}{x^{-12}} $$ (c) Simplify the following, leaving your answers with positive exponents: (1) $$ \left(x^{-3}ight)^{4} $$ (2) $$ \left(x^{2}ight)^{-5} $$ (3) $$ \left(x^{-2}ight)^{-2} $$ (4) $$ \left(-4 x^{-3}ight)^{3} $$ (5) $$ \left(-2 x^{3}ight)^{4} $$ (6) $$ \left(x^{4}ight)^{2}+\left(x^{2}ight)^{4} $$ (7) $$ 2\left(x^{4}ight)^{2}+4\left(x^{2}ight)^{4} $$ (8) $$ \left(-x^{3}ight)^{2} \cdot\left(-x^{2}ight)^{3} $$ (9) $$ 2\left(-x^{4}ight)^{2} 	imes 4\left(-x^{2}ight. $$ (10) $$ \left(3 x^{4}ight)^{2}-\left(2 x^{2}ight)^{4} $$ (11) $$ 3\left(-2 a^{3}ight)^{2}-2\left(-3 a^{2}ight)^{3} $$ (12) $$ \left(3 x^{4}ight)^{2} \cdot\left(2 x^{2}ight)^{4} $$ (13) $$ 3\left(-2 a^{3}ight)^{2} 	imes 2\left(-3 a^{2}ight)^{3} $$ (d) Simplify the following, leaving your answers with positive exponents: (1) $$ \left(\frac{a^{4}}{b^{6}}ight)^{2} $$ (2) $$ \left(\frac{10 x^{3} y}{5 x y^{2}}ight)^{3} $$ (3) $$ \left(\frac{-2 p^{4} q^{-2}}{4 p^{-6}}ight)^{3} $$ (4) $$ \left(\frac{3 x^{-5}}{9 x^{7}}ight)^{-2} $$ (5) $$ \left(\frac{2 a^{3} \cdot 4 a^{2}}{8\left(a^{-3}ight)^{2}}ight)^{2} $$ (6) $$ \left(\frac{x^{2}+x^{2}}{x^{2} 	imes x^{2}}ight)^{2} $$ $$ \frac{\left(x^{-3} y^{5}ight)^{2}}{x^{3} y^{-6}} $$ (81) $$ \frac{-3\left(a^{-1} b^{2}ight)^{-3} 	imes(a b)^{-5}}{\left(-3 b^{-5}ight)^{2}} $$ (9) $$ \frac{x+y}{x^{-1}+y^{-1}} $$

UpStudy AI · Accepted Answer

Let's simplify the expressions step by step as requested.

### Exercise 2

#### (a) Simplify the following:

1. **$$ 2 x^{3} + 2 x^{3} $$**

This is a straightforward addition of like terms:
   $$
   2 x^{3} + 2 x^{3} = 4 x^{3}
   $$

2. **$$ 2 x^{3} \times 2 x^{3} $$**

Here, we multiply the coefficients and add the exponents:
   $$
   2 x^{3} \times 2 x^{3} = 4 x^{6}
   $$

3. **$$ 2 x^{3} + 2 x^{3} $$** (same as the first)

$$
   2 x^{3} + 2 x^{3} = 4 x^{3}
   $$

4. **$$ 3 x^{3} \times 2 x^{3} + 6 x^{2} \times 2 x^{4} $$**

First, simplify each multiplication:
   $$
   3 x^{3} \times 2 x^{3} = 6 x^{6}
   $$
   $$
   6 x^{2} \times 2 x^{4} = 12 x^{6}
   $$
   Now, add the results:
   $$
   6 x^{6} + 12 x^{6} = 18 x^{6}
   $$

5. **$$ \left(3 x^{3} \times 2 x^{3}\right) + \left(3 x^{3} + 2 x^{3}\right) $$**

First, simplify the products:
   $$
   3 x^{3} \times 2 x^{3} = 6 x^{6}
   $$
   Now simplify the sum:
   $$
   3 x^{3} + 2 x^{3} = 5 x^{3}
   $$
   Finally, combine:
   $$
   6 x^{6} + 5 x^{3}
   $$

#### (b) Simplify the following, leaving your answer with positive exponents:

1. **$$ \frac{a^{6} b^{-8} c}{a^{-8} b^{6} c^{-3}} $$**

Apply the properties of exponents:
   $$
   = a^{6 - (-8)} b^{-8 - 6} c^{1 - (-3)} = a^{14} b^{-14} c^{4}
   $$
   Rewrite with positive exponents:
   $$
   = \frac{a^{14} c^{4}}{b^{14}}
   $$

2. **$$ \frac{x^{-12}}{x^{-7}} $$**

Simplifying gives:
   $$
   = x^{-12 - (-7)} = x^{-5} = \frac{1}{x^{5}}
   $$

3. **$$ \frac{x^{-7}}{x^{-12}} $$**

Simplifying gives:
   $$
   = x^{-7 - (-12)} = x^{5}
   $$

#### (c) Simplify the following, leaving your answers with positive exponents:

1. **$$ \left(x^{-3}\right)^{4} $$**

Apply the power of a power rule:
   $$
   = x^{-12} = \frac{1}{x^{12}}
   $$

2. **$$ \left(x^{2}\right)^{-5} $$**

Apply the power of a power rule:
   $$
   = x^{-10} = \frac{1}{x^{10}}
   $$

3. **$$ \left(x^{-2}\right)^{-2} $$**

Apply the power of a power rule:
   $$
   = x^{4}
   $$

4. **$$ \left(-4 x^{-3}\right)^{3} $$**

Distributing the exponent:
   $$
   = (-4)^{3} (x^{-3})^{3} = -64 x^{-9} = \frac{-64}{x^{9}}
   $$

5. **$$ \left(-2 x^{3}\right)^{4} $$**

Distributing the exponent:
   $$
   = (-2)^{4} (x^{3})^{4} = 16 x^{12}
   $$

6. **$$ \left(x^{4}\right)^{2} + \left(x^{2}\right)^{4} $$**

Simplifying each term:
   $$
   = x^{8} + x^{8} = 2 x^{8}
   $$

7. **$$ 2\left(x^{4}\right)^{2} + 4\left(x^{2}\right)^{4} $$**

Simplifying each term:
   $$
   = 2 x^{8} + 4 x^{8} = 6 x^{8}
   $$

8. **$$ \left(-x^{3}\right)^{2} \cdot \left(-x^{2}\right)^{3} $$**

Simplifying each term:
   $$
   = x^{6} \cdot (-x^{6}) = -x^{12}
   $$

9. **$$ 2\left(-x^{4}\right)^{2} \times 4\left(-x^{2}\right) $$**

Simplifying gives:
   $$
   = 2 \cdot 4 x^{8} \cdot (-x^{2}) = -8 x^{10}
   $$

10. **$$ \left(3 x^{4}\right)^{2} - \left(2 x^{2}\right)^{4} $$**

Simplifying gives:
    $$
    = 9 x^{8} - 16 x^{8} = -7 x^{8}
    $$

11. **$$ 3\left(-2 a^{3}\right)^{2} - 2\left(-3 a^{2}\right)^{3} $$**

Simplifying gives:
    $$
    = 3 \cdot 4 a^{6} - 2 \cdot (-27 a^{6}) = 12 a^{6} + 54 a^{6} = 66 a^{6}
    $$

12. **$$ \left(3 x^{4}\right)^{2} \cdot \left(2 x^{2}\right)^{4} $$**

Simplifying gives:
    $$
    = 9 x^{8} \cdot 16 x^{8} = 144 x^{16}
    $$

13. **$$ 3\left(-2 a^{3}\right)^{2} \times 2\left(-3 a^{2}\right)^{3
### Simplified Solutions for Exercise 2 #### (a) Simplify the following: 1. $$ 2 x^{3} + 2 x^{3} = 4 x^{3} $$ 2. $$ 2 x^{3} \times 2 x^{3} = 4 x^{6} $$ 3. $$ 2 x^{3} + 2 x^{3} = 4 x^{3} $$ 4. $$ 3 x^{3} \times 2 x^{3} + 6 x^{2} \times 2 x^{4} = 18 x^{6} $$ 5. $$ \left(3 x^{3} \times 2 x^{3}\right) + \left(3 x^{3} + 2 x^{3}\right) = 6 x^{6} + 5 x^{3} $$ #### (b) Simplify the following, leaving your answer with positive exponents: 1. $$ \frac{a^{6} b^{-8} c}{a^{-8} b^{6} c^{-3}} = \frac{a^{14} c^{4}}{b^{14}} $$ 2. $$ \frac{x^{-12}}{x^{-7}} = \frac{1}{x^{5}} $$ 3. $$ \frac{x^{-7}}{x^{-12}} = x^{5} $$ #### (c) Simplify the following, leaving your answers with positive exponents: 1. $$ \left(x^{-3}\right)^{4} = \frac{1}{x^{12}} $$ 2. $$ \left(x^{2}\right)^{-5} = \frac{1}{x^{10}} $$ 3. $$ \left(x^{-2}\right)^{-2} = x^{4} $$ 4. $$ \left(-4 x^{-3}\right)^{3} = \frac{-64}{x^{9}} $$ 5. $$ \left(-2 x^{3}\right)^{4} = 16 x^{12} $$ 6. $$ \left(x^{4}\right)^{2} + \left(x^{2}\right)^{4} = 2 x^{8} $$ 7. $$ 2\left(x^{4}\right)^{2} + 4\left(x^{2}\right)^{4} = 6 x^{8} $$ 8. $$ \left(-x^{3}\right)^{2} \cdot \left(-x^{2}\right)^{3} = -x^{12} $$ 9. $$ 2\left(-x^{4}\right)^{2} \times 4\left(-x^{2}\right) = -8 x^{10} $$ 10. $$ \left(3 x^{4}\right)^{2} - \left(2 x^{2}\right)^{4} = -7 x^{8} $$ 11. $$ 3\left(-2 a^{3}\right)^{2} - 2\left(-3 a^{2}\right)^{3} = 66 a^{6} $$ 12. $$ \left(3 x^{4}\right)^{2} \cdot \left(2 x^{2}\right)^{4} = 144 x^{16} $$ 13. $$ 3\left(-2 a^{3}\right)^{2} \times 2\left(-3 a^{2}\right)^{3} = 3 \cdot 4 a^{6} \times 2 \cdot (-27 a^{6}) = -162 a^{12} $$ #### (d) Simplify the following, leaving your answers with positive exponents: 1. $$ \left(\frac{a^{4}}{b^{6}}\right)^{2} = \frac{a^{8}}{b^{12}} $$ 2. $$ \left(\frac{10 x^{3} y}{5 x y^{2}}\right)^{3} = \frac{1000 x^{9} y^{3}}{125 x^{3} y^{6}} = \frac{8 x^{6}}{y^{3}} $$ 3. $$ \left(\frac{-2 p^{4} q^{-2}}{4 p^{-6}}\right)^{3} = \frac{(-2)^{3} p^{12} q^{-6}}{4^{3}} = \frac{-8 p^{12}}{64 q^{6}} = \frac{-p^{12}}{8 q^{6}} $$ 4. $$ \left(\frac{3 x^{-5}}{9 x^{7}}

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