\begin{tabular}{l} 15. \( 7 x^{4}-5 x^{4}= \) \\ \begin{tabular}{ll} 17. \( 6 b+7 b-10= \) & 16. \( 32 y+5 y= \) \\ 19. \( y+4+3(y+2)= \) & 20. \( 7 a^{2}-a^{2}+16= \) \\ 21. \( 3 y^{2}+3\left(4 y^{2}-2\right)=\square \) & 22. \( z^{2}+z+4 z^{3}+4 z^{2}= \) \\ 23. \( 0.5\left(x^{4}+3\right)+12=\square \) \\ 25. Justify Reasoning Determine whether \( 3 x+12+x \) is equivalent to \\ \( 4(3+x) \). Use properties of operations to justify your answer. \\ 24. \( 16+4 p)= \) \\ \hline\end{tabular} \\ \( \begin{array}{ll}\text { 26. Ted earns } \$ 13 \text { an hour at a theater. Last week he worked } h \text { hours at the } \\ \text { concession stand and three times as many hours at the ticket counter. } \\ \text { Write and simplify an expression for the amount he earned last week. }\end{array} \) \\ \hline\end{tabular}

Simplify the expression by following steps: - step0: Solution: \(16+4p\) Calculate or simplify the expression \( 3*y^2+3*(4*y^2-2) \). Simplify the expression by following steps: - step0: Solution: \(3y^{2}+3\left(4y^{2}-2\right)\) - step1: Expand the expression: \(3y^{2}+12y^{2}-6\) - step2: Add the terms: \(15y^{2}-6\) Calculate or simplify the expression \( 13*h+13*3*h \). Simplify the expression by following steps: - step0: Solution: \(13h+13\times 3h\) - step1: Multiply the terms: \(13h+39h\) - step2: Collect like terms: \(\left(13+39\right)h\) - step3: Add the numbers: \(52h\) Calculate or simplify the expression \( 3*x+12+x \). Simplify the expression by following steps: - step0: Solution: \(3x+12+x\) - step1: Add the terms: \(4x+12\) Calculate or simplify the expression \( z^2+z+4*z^3+4*z^2 \). Simplify the expression by following steps: - step0: Solution: \(z^{2}+z+4z^{3}+4z^{2}\) - step1: Add the terms: \(5z^{2}+z+4z^{3}\) Calculate or simplify the expression \( 4*(3+x) \). Simplify the expression by following steps: - step0: Solution: \(4\left(3+x\right)\) - step1: Rearrange the terms: \(12+4x\) Calculate or simplify the expression \( 32*y+5*y \). Simplify the expression by following steps: - step0: Solution: \(32y+5y\) - step1: Collect like terms: \(\left(32+5\right)y\) - step2: Add the numbers: \(37y\) Calculate or simplify the expression \( 7*x^4-5*x^4 \). Simplify the expression by following steps: - step0: Solution: \(7x^{4}-5x^{4}\) - step1: Collect like terms: \(\left(7-5\right)x^{4}\) - step2: Subtract the numbers: \(2x^{4}\) Calculate or simplify the expression \( 6*b+7*b-10 \). Simplify the expression by following steps: - step0: Solution: \(6b+7b-10\) - step1: Add the terms: \(13b-10\) Calculate or simplify the expression \( 0.5*(x^4+3)+12 \). Simplify the expression by following steps: - step0: Solution: \(0.5\left(x^{4}+3\right)+12\) - step1: Expand the expression: \(0.5x^{4}+1.5+12\) - step2: Add the numbers: \(0.5x^{4}+13.5\) Calculate or simplify the expression \( y+4+3*(y+2) \). Simplify the expression by following steps: - step0: Solution: \(y+4+3\left(y+2\right)\) - step1: Expand the expression: \(y+4+3y+6\) - step2: Add the terms: \(4y+10\) Calculate or simplify the expression \( 7*a^2-a^2+16 \). Simplify the expression by following steps: - step0: Solution: \(7a^{2}-a^{2}+16\) - step1: Subtract the terms: \(6a^{2}+16\) Let's solve each of the problems step by step: 1. **Problem 15: \( 7x^{4} - 5x^{4} \)** - Combine like terms: \[ 7x^{4} - 5x^{4} = (7 - 5)x^{4} = 2x^{4} \] - **Answer:** \( 2x^{4} \) 2. **Problem 17: \( 6b + 7b - 10 \)** - Combine like terms: \[ 6b + 7b = (6 + 7)b = 13b \] - Therefore, the expression simplifies to: \[ 13b - 10 \] - **Answer:** \( 13b - 10 \) 3. **Problem 16: \( 32y + 5y \)** - Combine like terms: \[ 32y + 5y = (32 + 5)y = 37y \] - **Answer:** \( 37y \) 4. **Problem 19: \( y + 4 + 3(y + 2) \)** - Distribute \( 3 \): \[ y + 4 + 3y + 6 = (1y + 3y) + (4 + 6) = 4y + 10 \] - **Answer:** \( 4y + 10 \) 5. **Problem 20: \( 7a^{2} - a^{2} + 16 \)** - Combine like terms: \[ 7a^{2} - a^{2} = (7 - 1)a^{2} = 6a^{2} \] - Therefore, the expression simplifies to: \[ 6a^{2} + 16 \] - **Answer:** \( 6a^{2} + 16 \) 6. **Problem 21: \( 3y^{2} + 3(4y^{2} - 2) \)** - Distribute \( 3 \): \[ 3y^{2} + 12y^{2} - 6 = (3y^{2} + 12y^{2}) - 6 = 15y^{2} - 6 \] - **Answer:** \( 15y^{2} - 6 \) 7. **Problem 22: \( z^{2} + z + 4z^{3} + 4z^{2} \)** - Combine like terms: \[ z^{2} + 4z^{2} + 4z^{3} = (1 + 4)z^{2} + 4z^{3} = 5z^{2} + 4z^{3} \] - **Answer:** \( 5z^{2} + 4z^{3} \) 8. **Problem 23: \( 0.5(x^{4} + 3) + 12 \)** - Distribute \( 0.5 \): \[ 0.5x^{4} + 1.5 + 12 = 0.5x^{4} + 13.5 \] - **Answer:** \( 0.5x^{4} + 13.5 \) 9. **Problem 24: \( 16 + 4p \)** - This expression is already simplified. - **Answer:** \( 16 + 4p \) 10. **Problem 25: Justify Reasoning** - For \( 3x + 12 + x \): \[ 3x + 12 + x = (3 + 1)x + 12 = 4x + 12 \] - For \( 4(3 + x) \): \[ 4(3 + x) = 12 + 4x \] - Both expressions simplify to \( 4x + 12 \) and \( 12 + 4x \), which are equivalent. - **Conclusion:** Yes, \( 3x + 12 + x \) is equivalent to \( 4(3 + x) \). 11. **Problem 26: Ted's Earnings** - Ted earns \( \$13 \) an hour. He worked \( h \) hours at the concession stand and \( 3h \) hours at the ticket counter. - Total earnings: \[ 13h + 13(3h) = 13h + 39h = 52h \] - **Answer:** \( 52h \) This concludes the solutions for all the problems presented.