Simplify the following expressions and express the result in the form of positive exponents: (3/7)^2 × (3/7)^3, (1/2)^5 + (1/2)^3, etc.
Solution
Let's go through the questions one by one to simplify and express them with positive exponents.
a) (73)2×(73)3
Apply the multiplication rule for exponents: (am×an=am+n).
=(73)2+3=(73)5
b) (21)5+(21)3
The bases are the same, but since they are added, simplify each term separately:
= \frac{1}{32} + \frac{4}{32} = \frac{5}{32}$$
### c) $$\left(\frac{4}{3}\right)^{-2}$$
- A negative exponent means reciprocal:
$$= \left(\frac{3}{4}\right)^2 = \frac{9}{16}$$
### d) $$\left(\frac{2}{3}\right)^2 \times \left(\frac{-3}{2}\right)^3$$
- Handle each term:
$$= \left(\frac{2}{3}\right)^2 = \frac{4}{9}$$
$$= \left(\frac{-3}{2}\right)^3 = \frac{-27}{8}$$
Now, multiply them:
$$= \frac{4}{9} \times \frac{-27}{8} = \frac{-108}{72} = -\frac{3}{2}$$
### e) $$\left(\frac{2}{5}\right)^{3 \times 2}$$
- First multiply the exponents:
$$= \left(\frac{2}{5}\right)^6 = \frac{64}{15625}$$
### f) $$\left(\frac{3}{4}\right)^2 \times \left(\frac{1}{4}\right)^{-2} \div \left(\frac{4}{3}\right)^3$$
- Handle each term:
$$= \left(\frac{3}{4}\right)^2 = \frac{9}{16}$$
$$= \left(\frac{1}{4}\right)^{-2} = 4^2 = 16$$
$$= \left(\frac{4}{3}\right)^3 = \frac{64}{27}$$
Now, multiply and divide:
$$= \frac{9}{16} \times 16 \div \frac{64}{27}
= 9 \div \frac{64}{27}
= 9 \times \frac{27}{64} = \frac{243}{64}$$
### g) $$\left(\frac{1}{2}\right)^2 \times \left(\frac{1}{2}\right)^3 \div \left(\frac{1}{2}\right)^{-5}$$
- Use exponent rules:
$$= \left(\frac{1}{2}\right)^{2+3} \div \left(\frac{1}{2}\right)^{-5}
= \left(\frac{1}{2}\right)^5 \times \left(\frac{1}{2}\right)^5
= \left(\frac{1}{2}\right)^{10} = \frac{1}{1024}$$
Let me know if you'd like more detailed explanations on any specific part! Here are some related questions:
1. What are the general rules for working with negative exponents?
2. How do you handle adding fractions with different denominators?
3. How can you simplify complex fractions when dividing?
4. What happens when you multiply or divide numbers with the same base but different exponents?
5. How do you convert between different forms of fractions raised to powers?
**Tip**: Always remember that a negative exponent means taking the reciprocal of the base!
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Math Problem Analysis
Mathematical Concepts
Exponentiation Fraction Operations Multiplication of Exponents Addition of Fractions Negative Exponents