Algebraic Expressions and Simplification - Quantitative Reasoning

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

   $x^6$. Raise a power to a power by multiplying exponents: $3 cdot 2 = 6$.

2. What is the simplified form of $\frac{3x}{4}-\frac{x}{8}$?

   $\frac{5x}{8}$. Subtract fractions with common denominator 8: $\frac{6x}{8} - \frac{x}{8} = \frac{5x}{8}$.

3. What is the simplified form of $\frac{x^2-4}{x^2+4x+4}$ for $x\ne -2$?

   $\frac{x-2}{x+2}$. Factor numerator as $(x-2)(x+2)$ and denominator as $(x+2)^2$, then cancel one $(x+2)$.

4. What property justifies rewriting $ab+ac$ as $a(b+c)$?

   Factoring using the distributive property. Factoring reverses the distributive property by extracting the common factor $a$ from $ab + ac$ to form $a(b + c)$.

5. What is the simplified form of $x^2-10x+25$?

   $(x-5)^2$. Factor as perfect square trinomial: coefficients fit $(x-5)^2 = x^2 - 10x + 25$.

6. What is the simplified form of $3x+5x$?

   $8x$. Combine like terms by adding the coefficients of $x$, resulting in $3 + 5 = 8$.

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

   $x^2-25$. Apply difference of squares formula: $x^2 - 5^2$.

8. What is the simplified form of $\frac{2}{3}x+\frac{1}{6}x$?

   $\frac{5}{6}x$. Add fractions with common denominator 6: $\frac{4}{6}x + \frac{1}{6}x = \frac{5}{6}x$.

9. What is the simplified form of $x^0$ for $x\ne 0$?

   $1$. Any non-zero number raised to the power of 0 equals 1 by definition.

10. What is the simplified form of $x^{-3}$ using only positive exponents?

    $\frac{1}{x^3}$. A negative exponent indicates a reciprocal with the exponent made positive.

11. What is the simplified form of $7a-2a+9a$?

    $14a$. Combine like terms by computing the net coefficient: $7 - 2 + 9 = 14$.

12. What is the simplified form of $-(x-6)+2(x+1)$?

    $x+8$. Distribute the negative and the 2, then combine like terms: $-x + 6 + 2x + 2 = x + 8$.

13. 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.

14. What is the simplified form of $\frac{6a^3b^2}{2ab^5}$ using only positive exponents?

    $\frac{3a^2}{b^3}$. Convert negative exponent to positive by moving $b^{-3}$ to the denominator as $b^3$.

15. What property justifies rewriting $a(b+c)$ as $ab+ac$?

    Distributive property. The distributive property states that multiplication distributes over addition, allowing $a(b+c)$ to expand to $ab + ac$.

16. What is the simplified form of $(x^{-2})(x^5)$ for $x\ne 0$?

    $x^3$. Multiply powers with the same base by adding exponents: $-2 + 5 = 3$.

17. What is the simplified form of $\frac{x^2+5x}{x}$ for $x\ne 0$?

    $x+5$. Factor $x$ from numerator and cancel with denominator: $\frac{x(x+5)}{x} = x+5$.

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

    $x^2+6x+9$. Expand using FOIL or square binomial formula: $x^2 + 2cdot xcdot 3 + 3^2$.

19. What is the simplified form of $\frac{1}{x^{-2}}$ for $x\ne 0$?

    $x^2$. The reciprocal of a negative exponent simplifies to the base with a positive exponent: $x^{-(-2)} = x^2$.

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

    $6x-12$. Distribute the 4 and combine like terms: $4x - 12 + 2x = 6x - 12$.

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

    $8x^6y^3$. Distribute the exponent to each factor: $2^3=8$, $(x^2)^3=x^6$, $y^3=y^3$.

22. What is the simplified form of $\frac{x^5}{x^2}$ for $x\ne 0$?

    $x^3$. Divide powers with the same base by subtracting exponents: $5 - 2 = 3$.

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