Problem: Simplify the given expressions involving monomials.
We will solve each problem step by step, using the rules of multiplying and dividing monomials:
1.
Multiplying Monomials:
- Multiply the coefficients.
- Add the exponents of like bases.
2.
Dividing Monomials:
- Divide the coefficients.
- Subtract the exponents of like bases in the denominator from those in the numerator.
---
####
Problem 1: $(2x^4)(3x^5)$
-
Step 1: Multiply the coefficients: $2 \cdot 3 = 6$.
-
Step 2: Add the exponents of $x$: $x^4 \cdot x^5 = x^{4+5} = x^9$.
-
Solution: $6x^9$.
####
Problem 2: $(-4x^3)(3x^8)$
-
Step 1: Multiply the coefficients: $-4 \cdot 3 = -12$.
-
Step 2: Add the exponents of $x$: $x^3 \cdot x^8 = x^{3+8} = x^{11}$.
-
Solution: $-12x^{11}$.
####
Problem 3: $(3x^8)(12x^3)$
-
Step 1: Multiply the coefficients: $3 \cdot 12 = 36$.
-
Step 2: Add the exponents of $x$: $x^8 \cdot x^3 = x^{8+3} = x^{11}$.
-
Solution: $36x^{11}$.
####
Problem 4: $(6x^2)(15x^5)$
-
Step 1: Multiply the coefficients: $6 \cdot 15 = 90$.
-
Step 2: Add the exponents of $x$: $x^2 \cdot x^5 = x^{2+5} = x^7$.
-
Solution: $90x^7$.
####
Problem 5: $(x^5)(2x^4)$
-
Step 1: Multiply the coefficients: $1 \cdot 2 = 2$.
-
Step 2: Add the exponents of $x$: $x^5 \cdot x^4 = x^{5+4} = x^9$.
-
Solution: $2x^9$.
####
Problem 6: $(10x^8)(3x^3)$
-
Step 1: Multiply the coefficients: $10 \cdot 3 = 30$.
-
Step 2: Add the exponents of $x$: $x^8 \cdot x^3 = x^{8+3} = x^{11}$.
-
Solution: $30x^{11}$.
---
####
Problem 7: $\frac{63x^9}{7x^7}$
-
Step 1: Divide the coefficients: $\frac{63}{7} = 9$.
-
Step 2: Subtract the exponents of $x$: $x^9 \div x^7 = x^{9-7} = x^2$.
-
Solution: $9x^2$.
####
Problem 8: $\frac{54x^3}{3x}$
-
Step 1: Divide the coefficients: $\frac{54}{3} = 18$.
-
Step 2: Subtract the exponents of $x$: $x^3 \div x = x^{3-1} = x^2$.
-
Solution: $18x^2$.
####
Problem 9: $\frac{26x^{10}}{2x^5}$
-
Step 1: Divide the coefficients: $\frac{26}{2} = 13$.
-
Step 2: Subtract the exponents of $x$: $x^{10} \div x^5 = x^{10-5} = x^5$.
-
Solution: $13x^5$.
####
Problem 10: $\frac{10x^8}{4x^3}$
-
Step 1: Divide the coefficients: $\frac{10}{4} = \frac{5}{2}$.
-
Step 2: Subtract the exponents of $x$: $x^8 \div x^3 = x^{8-3} = x^5$.
-
Solution: $\frac{5}{2}x^5$.
####
Problem 11: $\frac{20x^4}{30x^3}$
-
Step 1: Divide the coefficients: $\frac{20}{30} = \frac{2}{3}$.
-
Step 2: Subtract the exponents of $x$: $x^4 \div x^3 = x^{4-3} = x^1 = x$.
-
Solution: $\frac{2}{3}x$.
####
Problem 12: $\frac{18x^7}{3x^5}$
-
Step 1: Divide the coefficients: $\frac{18}{3} = 6$.
-
Step 2: Subtract the exponents of $x$: $x^7 \div x^5 = x^{7-5} = x^2$.
-
Solution: $6x^2$.
---
Final Answers:
\[
\boxed{
\begin{aligned}
1. & \ 6x^9 \\
2. & \ -12x^{11} \\
3. & \ 36x^{11} \\
4. & \ 90x^7 \\
5. & \ 2x^9 \\
6. & \ 30x^{11} \\
7. & \ 9x^2 \\
8. & \ 18x^2 \\
9. & \ 13x^5 \\
10. & \ \frac{5}{2}x^5 \\
11. & \ \frac{2}{3}x \\
12. & \ 6x^2
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of multiplying and dividing monomials worksheet.
