Question: 4. Simplify.
(a) (9pq^3)^2
(b) 6(m^2n^3)^3 \div 2m^3n
(c) 2pq^{-1} \times (3p^2q^{-3})^2 \div 6p^3q^{-5}
5. (a) Given that 3^{2x - 1} = 27, calculate the value of x.
(b) Simplify (g^2)^2 \times (4fg^2)^3 \div f^2g^7.
6. Simplify.
(a) 3(9^{2x + 1})
(b) 4[2^{2m} \times (2^{1/3})^{-3}]
7. Given that 16 \times 2^p = 2^9, find the value of p.
1. **Problem 4: Simplify each expression.**
(a) Simplify $$(9pq^3)^2$$
- Use the power of a product rule: $$(ab)^n = a^n b^n$$
- So, $$(9pq^3)^2 = 9^2 \times p^2 \times (q^3)^2$$
- Calculate powers: $$9^2 = 81$$ and $$(q^3)^2 = q^{3 \times 2} = q^6$$
- Final expression: $$81p^2q^6$$
(b) Simplify $$6(m^2n^3)^3 \div 2m^3n$$
- First, expand the power: $$(m^2n^3)^3 = m^{2 \times 3} n^{3 \times 3} = m^6 n^9$$
- Substitute back: $$6m^6 n^9 \div 2m^3 n$$
- Divide coefficients: $$\frac{6}{2} = 3$$
- Divide variables using quotient rule: $$m^{6-3} = m^3$$ and $$n^{9-1} = n^8$$
- Final expression: $$3m^3 n^8$$
(c) Simplify $$2pq^{-1} \times (3p^2q^{-3})^2 \div 6p^3q^{-5}$$
- Expand the power: $$(3p^2q^{-3})^2 = 3^2 p^{2 \times 2} q^{-3 \times 2} = 9 p^4 q^{-6}$$
- Multiply numerator terms: $$2pq^{-1} \times 9 p^4 q^{-6} = 18 p^{1+4} q^{-1 + (-6)} = 18 p^5 q^{-7}$$
- Divide by denominator: $$\frac{18 p^5 q^{-7}}{6 p^3 q^{-5}}$$
- Divide coefficients: $$\frac{18}{6} = 3$$
- Divide variables: $$p^{5-3} = p^2$$ and $$q^{-7 - (-5)} = q^{-7 + 5} = q^{-2}$$
- Final expression: $$3 p^2 q^{-2}$$
2. **Problem 5: Exponent equations and simplification.**
(a) Given $$3^{2x - 1} = 27$$, find $$x$$.
- Express 27 as a power of 3: $$27 = 3^3$$
- So, $$3^{2x - 1} = 3^3$$
- Since bases are equal, exponents must be equal: $$2x - 1 = 3$$
- Solve for $$x$$: $$2x = 4 \Rightarrow x = 2$$
(b) Simplify $$(g^2)^2 \times (4fg^2)^3 \div f^2 g^7$$
- Expand powers: $$(g^2)^2 = g^{2 \times 2} = g^4$$
- Expand $$(4fg^2)^3 = 4^3 f^3 (g^2)^3 = 64 f^3 g^6$$
- Multiply numerator: $$g^4 \times 64 f^3 g^6 = 64 f^3 g^{4+6} = 64 f^3 g^{10}$$
- Divide by denominator: $$\frac{64 f^3 g^{10}}{f^2 g^7} = 64 f^{3-2} g^{10-7} = 64 f g^3$$
3. **Problem 6: Simplify expressions with powers.**
(a) Simplify $$3(9^{2x + 1})$$
- Express 9 as $$3^2$$: $$9^{2x + 1} = (3^2)^{2x + 1} = 3^{2(2x + 1)} = 3^{4x + 2}$$
- Multiply by 3: $$3 \times 3^{4x + 2} = 3^{1 + 4x + 2} = 3^{4x + 3}$$
(b) Simplify $$4[2^{2m} \times (2^{1/3})^{-3}]$$
- Simplify inside brackets: $$(2^{1/3})^{-3} = 2^{(1/3) \times (-3)} = 2^{-1} = \frac{1}{2}$$
- Multiply powers of 2: $$2^{2m} \times 2^{-1} = 2^{2m - 1}$$
- Multiply by 4: $$4 \times 2^{2m - 1} = 2^2 \times 2^{2m - 1} = 2^{2 + 2m - 1} = 2^{2m + 1}$$
4. **Problem 7: Solve for $$p$$ in $$16 \times 2^p = 2^9$$**
- Express 16 as a power of 2: $$16 = 2^4$$
- Substitute: $$2^4 \times 2^p = 2^9$$
- Combine powers: $$2^{4 + p} = 2^9$$
- Equate exponents: $$4 + p = 9$$
- Solve for $$p$$: $$p = 5$$