Let's solve each of these mathematical expressions step-by-step.
### Question 1: Simplify [tex]\(\frac{24 c^5}{18 c^3}\)[/tex]
1. Divide the coefficients: [tex]\(\frac{24}{18} = \frac{24 \div 6}{18 \div 6} = \frac{4}{3}\)[/tex].
2. Subtract the exponents of [tex]\(c\)[/tex]: When dividing like bases, subtract the exponents: [tex]\(c^{5-3} = c^2\)[/tex].
So, the simplification of [tex]\(\frac{24 c^5}{18 c^3}\)[/tex] is:
[tex]\[ \frac{4}{3} c^2. \][/tex]
### Question 2: Simplify [tex]\(\frac{72 a^9 b^8}{24 a^7 b^5}\)[/tex]
1. Divide the coefficients: [tex]\(\frac{72}{24} = \frac{72 \div 24}{24 \div 24} = 3\)[/tex].
2. Subtract the exponents of [tex]\(a\)[/tex]: [tex]\(a^{9-7} = a^2\)[/tex].
3. Subtract the exponents of [tex]\(b\)[/tex]: [tex]\(b^{8-5} = b^3\)[/tex].
So, the simplification of [tex]\(\frac{72 a^9 b^8}{24 a^7 b^5}\)[/tex] is:
[tex]\[ 3 a^2 b^3. \][/tex]
### Question 3: Simplify [tex]\(10 a^2 \times 4 d^2 e^2\)[/tex]
1. Multiply the coefficients: [tex]\(10 \times 4 = 40\)[/tex].
2. Combine the variables: [tex]\(a^2\)[/tex], [tex]\(d^2\)[/tex], and [tex]\(e^2\)[/tex] are already in their simplest form since we are multiplying different bases.
So, the simplification of [tex]\(10 a^2 \times 4 d^2 e^2\)[/tex] is:
[tex]\[ 40 a^2 d^2 e^2. \][/tex]
### Question 4: Simplify [tex]\((4 d)^4\)[/tex]
1. Raise both the coefficient and the variable to the power of 4:
- Coefficient: [tex]\(4^4 = 4 \times 4 \times 4 \times 4 = 256\)[/tex].
- Variable: [tex]\((d^1)^4 = d^{1 \times 4} = d^4\)[/tex].
So, the simplification of [tex]\((4 d)^4\)[/tex] is:
[tex]\[ 256 d^4. \][/tex]
### Final Solutions
1. [tex]\(\frac{24 c^5}{18 c^3} = \frac{4}{3} c^2\)[/tex]
2. [tex]\(\frac{72 a^9 b^8}{24 a^7 b^5} = 3 a^2 b^3\)[/tex]
3. [tex]\(10 a^2 \times 4 d^2 e^2 = 40 a^2 d^2 e^2\)[/tex]
4. [tex]\((4 d)^4 = 256 d^4\)[/tex]
