Question
- Exponents and roots are not distributive oras \( (a+b)^{n} \neq a^{n}+b^{n}, \quad(a-b)^{n} \neq a^{n}-b^{n}, \quad \sqrt[n]{a+b} \neq \sqrt[n]{a}+\sqrt[n]{b} \) and \( \sqrt[n]{a-b} \neq \sqrt[n]{a} \). EXERCISE 2 (a) Simplify the following: (2) \( 2 x^{3}+2 x^{3} \) (1) \( 2 x^{3} \times 2 x^{3} \) (2) \( 2 x^{3}+2 x^{3} \) (4) \( 3 x^{3} \times 2 x^{3}+6 x^{2} \times 2 x^{4} \) (3) \( \left(3 x^{3} \times 2 x^{3}\right)+\left(3 x^{3}+2 x^{3}\right) \) (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) \( \left(x^{-3}\right)^{4} \) (2) \( \left(x^{2}\right)^{-5} \) (3) \( \left(x^{-2}\right)^{-2} \) (4) \( \left(-4 x^{-3}\right)^{3} \) (5) \( \left(-2 x^{3}\right)^{4} \) (6) \( \left(x^{4}\right)^{2}+\left(x^{2}\right)^{4} \) (7) \( 2\left(x^{4}\right)^{2}+4\left(x^{2}\right)^{4} \) (8) \( \left(-x^{3}\right)^{2} \cdot\left(-x^{2}\right)^{3} \) (9) \( 2\left(-x^{4}\right)^{2} \times 4\left(-x^{2}\right. \) (10) \( \left(3 x^{4}\right)^{2}-\left(2 x^{2}\right)^{4} \) (11) \( 3\left(-2 a^{3}\right)^{2}-2\left(-3 a^{2}\right)^{3} \) (12) \( \left(3 x^{4}\right)^{2} \cdot\left(2 x^{2}\right)^{4} \) (13) \( 3\left(-2 a^{3}\right)^{2} \times 2\left(-3 a^{2}\right)^{3} \) (d) Simplify the following, leaving your answers with positive exponents: (1) \( \left(\frac{a^{4}}{b^{6}}\right)^{2} \) (2) \( \left(\frac{10 x^{3} y}{5 x y^{2}}\right)^{3} \) (3) \( \left(\frac{-2 p^{4} q^{-2}}{4 p^{-6}}\right)^{3} \) (4) \( \left(\frac{3 x^{-5}}{9 x^{7}}\right)^{-2} \) (5) \( \left(\frac{2 a^{3} \cdot 4 a^{2}}{8\left(a^{-3}\right)^{2}}\right)^{2} \) (6) \( \left(\frac{x^{2}+x^{2}}{x^{2} \times x^{2}}\right)^{2} \) \( \frac{\left(x^{-3} y^{5}\right)^{2}}{x^{3} y^{-6}} \) (81) \( \frac{-3\left(a^{-1} b^{2}\right)^{-3} \times(a b)^{-5}}{\left(-3 b^{-5}\right)^{2}} \) (9) \( \frac{x+y}{x^{-1}+y^{-1}} \)
Ask by Barnett Spencer. in South Africa
Feb 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
### 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}}
Solution
Sign in to Unlock Answers for Free!
A Learning Platform Trusted by Millions of Real Students and Teachers.
Unlock Answered by UpStudy AI and reviewed by a Professional Tutor
Like
Bonus Knowledge
First additional information: When you encounter polynomials, simple addition or multiplication can simplify expressions significantly! For example, when simplifying \(2x^3 + 2x^3\), you can simply combine the like terms to make it \(4x^3\). This not only makes the expression cleaner but can help you solve more complex problems faster. Remember, combining like terms is your friend! Second additional information: Exponents can be tricky, but the key is to remember that multiplying exponents means you add their powers, while dividing means you subtract. For example, in \( \frac{x^{-12}}{x^{-7}} \), you subtract the exponents to get \(x^{-12 - (-7)} = x^{-5}\), which you can then represent as \(\frac{1}{x^5}\) to ensure positive exponents. Keep this in mind, and you'll be zipping through exponent problems in no time!
