Here are the step-by-step solutions for each problem on the worksheet.
1. $3(x + 4)$
* Multiply 3 by $x$: $3 \cdot x = 3x$
* Multiply 3 by 4: $3 \cdot 4 = 12$
* Combine them:
$3x + 12$
2. $8(x - 7)$
* Multiply 8 by $x$: $8 \cdot x = 8x$
* Multiply 8 by -7: $8 \cdot -7 = -56$
* Combine them:
$8x - 56$
3. $-2(x - 3)$
* Multiply -2 by $x$: $-2 \cdot x = -2x$
* Multiply -2 by -3: $-2 \cdot -3 = +6$ (negative times negative is positive)
* Combine them:
$-2x + 6$
4. $5(2a - 4b)$
* Multiply 5 by $2a$: $5 \cdot 2a = 10a$
* Multiply 5 by $-4b$: $5 \cdot -4b = -20b$
* Combine them:
$10a - 20b$
5. $7(x^2 + 4x - 3)$
* Multiply 7 by $x^2$: $7x^2$
* Multiply 7 by $4x$: $28x$
* Multiply 7 by -3: $-21$
* Combine them:
$7x^2 + 28x - 21$
6. $c^2d^3(5cd^7 - 3c^3d^2 - 4d^3)$
* First term: $c^2d^3 \cdot 5cd^7$. Add exponents for like bases ($c^{2+1}d^{3+7}$). Result: $5c^3d^{10}$
* Second term: $c^2d^3 \cdot -3c^3d^2$. Add exponents ($c^{2+3}d^{3+2}$). Result: $-3c^5d^5$
* Third term: $c^2d^3 \cdot -4d^3$. Add exponents for $d$ ($d^{3+3}$), keep $c^2$. Result: $-4c^2d^6$
* Combine them:
$5c^3d^{10} - 3c^5d^5 - 4c^2d^6$
7. $-5x(-3x + 7)$
* Multiply $-5x$ by $-3x$: $(-5 \cdot -3)(x \cdot x) = 15x^2$
* Multiply $-5x$ by 7: $-35x$
* Combine them:
$15x^2 - 35x$
8. $3(y - 2) + 2y = 4y + 14$
* Distribute the 3: $3y - 6 + 2y = 4y + 14$
* Combine like terms on the left ($3y + 2y$): $5y - 6 = 4y + 14$
* Subtract $4y$ from both sides: $y - 6 = 14$
* Add 6 to both sides:
$y = 20$
9. $2x^2y(3xy - 2y)$
* First term: $2x^2y \cdot 3xy$. Multiply coefficients ($2 \cdot 3=6$). Add exponents for $x$ ($2+1=3$) and $y$ ($1+1=2$). Result: $6x^3y^2$
* Second term: $2x^2y \cdot -2y$. Multiply coefficients ($2 \cdot -2=-4$). Keep $x^2$. Add exponents for $y$ ($1+1=2$). Result: $-4x^2y^2$
* Combine them:
$6x^3y^2 - 4x^2y^2$
10. $d(d + 3) - d(d - 4) = 9d - 16$
* Distribute the first $d$: $d^2 + 3d$
* Distribute the second $-d$: $-d^2 + 4d$ (be careful with signs: $-d \cdot -4 = +4d$)
* Combine the left side: $(d^2 - d^2) + (3d + 4d) = 7d$
* Equation becomes: $7d = 9d - 16$
* Subtract $9d$ from both sides: $-2d = -16$
* Divide by -2:
$d = 8$
Final Answer:
1. $3x + 12$
2. $8x - 56$
3. $-2x + 6$
4. $10a - 20b$
5. $7x^2 + 28x - 21$
6. $5c^3d^{10} - 3c^5d^5 - 4c^2d^6$
7. $15x^2 - 35x$
8. $y = 20$
9. $6x^3y^2 - 4x^2y^2$
10. $d = 8$
Parent Tip: Review the logic above to help your child master the concept of simplifying monomials worksheet.
