Simplify Algebraic Expressions - International Brotherhood of Electrical Workers Exam IBEW

1. What is the simplified form of $\frac{a^2b^3}{ab}$ for $a \ne 0$ and $b \ne 0$?

   $ab^2$. Divide coefficients $a^2 / a = a$, and for $b$: $b^3 / b = b^2$.

2. What is the simplified form of $-3(2x - 5)$?

   $-6x + 15$. Distribute the -3 to each term inside the parentheses: $-3 cdot 2x = -6x$ and $-3 cdot (-5) = 15$.

3. Find and correct the simplification error: $2x + 3x^2 = 5x^3$.

   Correct: $2x + 3x^2$ cannot be combined. $2x$ and $3x^2$ are unlike terms with different exponents, so they cannot be added, and the given result uses incorrect exponent and coefficient.

4. What is the simplified form of $x(x + 7)$?

   $x^2 + 7x$. Distribute $x$ to each term: $x cdot x = x^2$ and $x cdot 7 = 7x$.

5. What is the simplified form of $5(x + 2)$?

   $5x + 10$. Apply the distributive property by multiplying each term inside the parentheses by 5.

6. What is the simplified form of $\frac{x^2 - 9}{x - 3}$ for $x \ne 3$?

   $x + 3$. Factor numerator as $(x-3)(x+3)$ and cancel common factor $(x-3)$ with denominator.

7. What is the simplified form of $3x - (2x - 5)$?

   $x + 5$. Distribute the negative: $3x - 2x + 5$, then combine like terms: $3x - 2x = x$.

8. What is the simplified form of $(x + 3)(x + 2)$?

   $x^2 + 5x + 6$. Use FOIL method: first $x cdot x = x^2$, outer $x cdot 2 = 2x$, inner $3 cdot x = 3x$, last $3 cdot 2 = 6$, then combine $2x + 3x = 5x$.

9. What is the simplified form of $x^3 \cdot x^5$?

   $x^8$. When multiplying powers with the same base, add the exponents: $3 + 5 = 8$.

10. What does it mean to combine like terms in an algebraic expression?

    Add or subtract terms with identical variable parts (same variables and exponents). Combining like terms means adding or subtracting coefficients of terms that have the exact same variables raised to the same powers.

11. What is the simplified form of $2x^2y - 5xy^2 + 3x^2y$?

    $5x^2y - 5xy^2$. Combine like terms: $2x^2 y + 3x^2 y = 5x^2 y$, while $-5xy^2$ remains separate.

12. What is the simplified form of $\frac{4x}{8}$?

    $\frac{x}{2}$. Simplify the fraction by dividing numerator and denominator by 4: $4/8 = 1/2$.

13. What is the simplified form of $\frac{x^2 + 2x}{x}$ for $x \ne 0$?

    $x + 2$. Factor $x$ from numerator and cancel with denominator: $x(x + 2)/x$.

14. What is the simplified form of $2(x - 4) + 3(x + 1)$?

    $5x - 5$. Distribute to get $2x - 8 + 3x + 3$, then combine like terms: $2x + 3x = 5x$ and $-8 + 3 = -5$.

15. What is the simplified form of $(x - 4)(x + 4)$?

    $x^2 - 16$. Apply difference of squares formula: $(x - a)(x + a) = x^2 - a^2$, where $a = 4$.

16. What is the simplified form of $\frac{x^7}{x^2}$ for $x \ne 0$?

    $x^5$. When dividing powers with the same base, subtract the exponents: $7 - 2 = 5$.

17. What is the simplified form of $12a - 5a + a$?

    $8a$. Combine like terms by computing the coefficients: $12 - 5 + 1 = 8$.

18. What is the simplified form of $-(-x + 6)$?

    $x - 6$. Distribute the negative sign: $-1 cdot (-x) = x$ and $-1 cdot 6 = -6$.

19. What is the simplified form of $(x + 5)^2$?

    $x^2 + 10x + 25$. Expand using $(a + b)^2 = a^2 + 2ab + b^2$, with $a = x$ and $b = 5$.

20. What is the simplified form of $6y - 2 + 3y + 5$?

    $9y + 3$. Group and combine like terms: $6y + 3y = 9y$ and $-2 + 5 = 3$.

21. What is the simplified form of $4x^2 + 9x^2 - 2x^2$?

    $11x^2$. Combine like terms by adding and subtracting coefficients: $4 + 9 - 2 = 11$.

22. What is the simplified form of $7x + 3x$?

    $10x$. Combine like terms by adding their coefficients: $7 + 3 = 10$.

23. What is the simplified form of $(2x)^3$?

    $8x^3$. Raise both coefficient and variable to the power: $2^3 = 8$ and $(x)^3 = x^3$.

24. Which terms are like terms: $3x^2$, $-5x^2$, $7x$, $x^2$?

    $3x^2$, $-5x^2$, and $x^2$ are like terms. Like terms share identical variables and exponents, so the $x^2$ terms can be combined while $7x$ differs in exponent.

25. What is the simplified form of $\frac{6x^3y}{2xy}$ for $x \ne 0$ and $y \ne 0$?

    $3x^2$. Divide coefficients $6/2 = 3$, subtract exponents for $x$: $3-1=2$, and cancel $y$: $1-1=0$.

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