Apply Properties of Integer Exponents - 8th Grade Math

1. What is the value of $a^0$ for $a \ne 0$?

   $a^0 = 1$. Any nonzero number to the zero power equals 1.

2. What is $(2^3)^4$ written as a single power of $2$?

   $2^{12}$. Apply power of a power rule: $2^{3 \cdot 4} = 2^{12}$.

3. What is the value of $10^{-2}$?

   $\frac{1}{100}$. $10^{-2} = \frac{1}{10^2} = \frac{1}{100}$.

4. What is the value of $\left(\frac{a}{b}\right)^{-n}$ for $a,b \ne 0$?

   $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$. A fraction with negative exponent equals its reciprocal with positive exponent.

5. Find and correct the error: $\frac{a^6}{a^2} = a^3$.

   Correct: $\frac{a^6}{a^2} = a^4$. Quotient rule subtracts exponents: $6-2=4$, not $6÷2=3$.

6. What is the simplified form of $\left(\frac{3}{5}\right)^3$?

   $\frac{3^3}{5^3}$. Using power of a quotient rule to distribute the exponent.

7. What is the zero exponent rule: $a^0 =$ ? (Assume $a \ne 0$)

   $1$. Any nonzero number raised to the power of zero equals one.

8. What is $\left(\frac{3x^{-2}}{6x^3}\right)$ simplified using positive exponents? (Assume $x \ne 0$)

   $\frac{1}{2x^5}$. Simplify coefficients and exponents: $\frac{3}{6} \cdot x^{-2-3} = \frac{1}{2}x^{-5}$.

9. State the negative exponent rule for $a^{-n}$ where $a \ne 0$ and $n$ is an integer.

   $a^{-n} = \frac{1}{a^n}$. A negative exponent means reciprocal with positive exponent.

10. What is the simplified form of $3^4 \cdot 3^{-6}$?

    $3^{-2}$. Using product rule: $3^{4+(-6)} = 3^{-2}$.

11. What is $\left(\frac{3}{5}\right)^2$ written as a fraction with exponents removed?

    $\frac{9}{25}$. $\frac{3^2}{5^2} = \frac{9}{25}$.

12. What is the value of $(2 \cdot 7)^3$ written as a product of powers?

    $2^3 \cdot 7^3$. Power of a product: $(ab)^n = a^n b^n$.

13. What is the value of $\frac{a^m}{a^m}$ for $a \ne 0$?

    $\frac{a^m}{a^m} = 1$. Any nonzero expression divided by itself equals 1.

14. State the power of a quotient rule for $\left(\frac{a}{b}\right)^n$ where $b \ne 0$.

    $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. A power distributes over division.

15. What is $\frac{2^{-4}}{2^{-1}}$ written as a single power of $2$?

    $2^{-3}$. Apply quotient rule: $2^{-4-(-1)} = 2^{-3}$.

16. What is $10^{-2}$ written as a fraction in simplest form?

    $\frac{1}{100}$. $10^{-2} = \frac{1}{10^2} = \frac{1}{100}$.

17. What is the value of $2^{-3}$?

    $\frac{1}{8}$. $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

18. State the negative exponent rule for $a^{-n}$ where $a \ne 0$ and $n$ is a positive integer.

    $a^{-n} = \frac{1}{a^n}$. A negative exponent means reciprocal with positive exponent.

19. What is the power of a product rule: $(ab)^n =$ ?

    $a^n b^n$. When raising a product to a power, apply the power to each factor.

20. What is the product rule for exponents: $a^m \cdot a^n =$ ?

    $a^{m+n}$. When multiplying powers with the same base, add the exponents.

21. State the quotient of powers rule for the same base $\frac{a^m}{a^n}$ where $a \ne 0$.

    $\frac{a^m}{a^n} = a^{m-n}$. When dividing powers with the same base, subtract the exponents.

22. What is $(5^2)^3$ written as a single power of $5$?

    $5^6$. Power of a power: $5^{2 \times 3} = 5^6$.

23. What is the value of $\frac{2^{-4}}{2^{-1}}$ expressed with a positive exponent?

    $\frac{1}{2^3}$. Quotient rule: $\frac{2^{-4}}{2^{-1}} = 2^{-4-(-1)} = 2^{-3} = \frac{1}{2^3}$.

24. What is $\left(\frac{3}{5}\right)^2$ written as a single fraction?

    $\frac{9}{25}$. Apply power of a quotient: $\frac{3^2}{5^2} = \frac{9}{25}$.

25. What is the value of $10^{-3}$ expressed as a fraction?

    $\frac{1}{10^3}$. Negative exponent rule: $a^{-n} = \frac{1}{a^n}$.

26. What is the value of $2^3 \cdot 2^5$?

    $2^8$. Product rule: $2^3 \cdot 2^5 = 2^{3+5} = 2^8$.

27. What is the value of $1^{-5}$?

    $1$. $1$ raised to any power equals $1$.

28. What is the simplified form of $\frac{x^{-2}}{x^5}$?

    $x^{-7}$. Using quotient rule: $x^{-2-5} = x^{-7}$.

29. What is the value of $3^2 \cdot 3^{-5}$ expressed with a positive exponent?

    $\frac{1}{3^3}$. Product rule gives $3^{2+(-5)} = 3^{-3} = \frac{1}{3^3}$.

30. State the product of powers rule for the same base $a^m \cdot a^n$.

    $a^m \cdot a^n = a^{m+n}$. When multiplying powers with the same base, add the exponents.

31. What is the value of $3^{-3}$ written as a fraction in simplest form?

    $\frac{1}{27}$. $3^{-3} = \frac{1}{3^3} = \frac{1}{27}$.

32. What is $3^4 \cdot 3^{-6}$ written as a single power of $3$?

    $3^{-2}$. Apply product rule: $3^{4+(-6)} = 3^{-2}$.

33. Find the simplified form of $\left(\frac{x^3 y^{-2}}{x^{-1}}\right)$ using exponent rules.

    $x^4 y^{-2}$. Simplify: $\frac{x^3 y^{-2}}{x^{-1}} = x^{3-(-1)} y^{-2} = x^4 y^{-2}$.

34. What is the quotient rule for exponents: $\frac{a^m}{a^n} =$ ? (Assume $a \ne 0$)

    $a^{m-n}$. When dividing powers with the same base, subtract the exponents.

35. What is the value of $3^2 \cdot 3^{-5}$ written as a single power of $3$?

    $3^{-3}$. Product rule: $3^{2+(-5)} = 3^{-3}$.

36. What is $\frac{4^{-1}}{4^2}$ written as a single power of $4$?

    $4^{-3}$. Quotient rule: $4^{-1-2} = 4^{-3}$.

37. State the power of a product rule for $ $(ab)^n$ $.

    $ $(ab)^n$ = $a^n$ $b^n$ $. A power distributes over multiplication.

38. State the negative exponent rule for $a^{-n}$ when $a \ne 0$.

    $a^{-n} = \frac{1}{a^n}$. A negative exponent means take the reciprocal with the positive exponent.

39. What is $2^{-1}$ written as a fraction in simplest form?

    $\frac{1}{2}$. $2^{-1} = \frac{1}{2^1} = \frac{1}{2}$.

40. What is the value of $a^0$ for any nonzero integer $a$?

    $a^0 = 1$. Any nonzero number to the zero power equals 1.

41. State the power of a product rule using $(ab)^n$.

    $(ab)^n = a^n b^n$. When raising a product to a power, apply the power to each factor.

42. What is the value of $(-1)^{13}$?

    $-1$. $(-1)$ to an odd power equals $-1$.

43. What is $\left(\frac{a}{b}\right)^{-n}$ rewritten with positive exponents? (Assume $a,b \ne 0$)

    $\left(\frac{b}{a}\right)^n$. Negative exponent on a fraction flips it and makes the exponent positive.

44. What is $7^{-3}$ written as a fraction with a positive exponent?

    $\frac{1}{7^3}$. Negative exponent rule: $7^{-3} = \frac{1}{7^3}$.

45. What is the simplified form of $\left(\frac{2}{3}\right)^{-2}$?

    $\left(\frac{3}{2}\right)^2$. Negative exponent flips the fraction: $\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2$.

46. State the product of powers rule for the same base using $a^m \cdot a^n$.

    $a^m \cdot a^n = a^{m+n}$. When multiplying powers with the same base, add the exponents.

47. What is $\frac{(2^5)(2^{-2})}{2^4}$ simplified as a single power of $2$?

    $2^{-1}$. Simplify numerator first: $\frac{2^3}{2^4} = 2^{-1}$.

48. Find and correct the error: $2^3 \cdot 2^4 = 2^{12}$.

    Correct: $2^3 \cdot 2^4 = 2^7$. Product rule adds exponents: $3+4=7$, not $3 \times 4=12$.

49. What is the negative exponent rule: $a^{-n} =$ ? (Assume $a \ne 0$)

    $\frac{1}{a^n}$. A negative exponent means take the reciprocal with positive exponent.

50. What is $\left(\frac{x^3}{x^{-2}}\right)$ simplified as a single power of $x$? (Assume $x \ne 0$)

    $x^5$. Apply quotient rule: $x^{3-(-2)} = x^5$.

51. Identify the equivalent expression for $\frac{a^7}{a^{10}}$ using integer exponent rules.

    $a^{-3}$. Quotient rule: $\frac{a^7}{a^{10}} = a^{7-10} = a^{-3}$.

52. What is $\frac{5^7}{5^3}$ written as a single power of $5$?

    $5^4$. Apply quotient rule: $5^{7-3} = 5^4$.

53. What is the simplified form of $(4 \cdot 7)^2$?

    $4^2 \cdot 7^2$. Using power of a product rule: $(4 \cdot 7)^2 = 4^2 \cdot 7^2$.

54. State the quotient of powers rule for the same base using $\frac{a^m}{a^n}$.

    $\frac{a^m}{a^n} = a^{m-n}$. When dividing powers with the same base, subtract the exponents.

55. What is $\left(\frac{2^3 \cdot 5^{-1}}{2^{-2} \cdot 5^2}\right)$ simplified using positive exponents?

    $\frac{2^5}{5^3}$. Apply quotient rule to each base: $2^{3-(-2)} \cdot 5^{-1-2} = 2^5 \cdot 5^{-3}$.

56. What is the value of $(-1)^{12}$?

    $1$. $(-1)$ to an even power equals $1$.

57. Find the equivalent expression for $\frac{(x^3)^2}{x^4}$ as a single power of $x$.

    $x^2$. $(x^3)^2 = x^6$, then $\frac{x^6}{x^4} = x^{6-4} = x^2$.

58. What is $4^{-3}$ written as a fraction in simplest form?

    $\frac{1}{64}$. $4^{-3} = \frac{1}{4^3} = \frac{1}{64}$.

59. What is the simplified form of $3^2 \cdot 3^{-5}$ written with a positive exponent?

    $\frac{1}{3^3}$. Product rule gives $3^{2+(-5)} = 3^{-3} = \frac{1}{3^3}$.

60. What is $\frac{2^7}{2^3}$ written as a single power of $2$?

    $2^4$. Quotient rule: $2^{7-3} = 2^4$.

61. What is the value of $\left(\frac{4}{5}\right)^2$ written as a quotient of powers?

    $\frac{4^2}{5^2}$. Power of a quotient: $(\frac{a}{b})^n = \frac{a^n}{b^n}$

62. What is $(2 \cdot 3)^4$ written using powers of $2$ and $3$?

    $2^4 \cdot 3^4$. Power of a product: distribute the exponent 4 to both factors.

63. What is the power of a quotient rule: $\left(\frac{a}{b}\right)^n =$ ? (Assume $b \ne 0$)

    $\frac{a^n}{b^n}$. When raising a quotient to a power, apply the power to both numerator and denominator.

64. What is the simplified form of $\frac{5^7}{5^3}$?

    $5^4$. Using quotient rule: $5^{7-3} = 5^4$.

65. State the power of a quotient rule using $\left(\frac{a}{b}\right)^n$.

    $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. When raising a quotient to a power, apply the power to both numerator and denominator.

66. What is $\left(\frac{a^2 b^{-1}}{a^{-3} b^4}\right)$ simplified using positive exponents?

    $\frac{a^5}{b^5}$. Simplify by subtracting exponents: $a^{2-(-3)} \cdot b^{-1-4} = a^5b^{-5}$.

67. What is the simplified form of $(2^3)^4$?

    $2^{12}$. Using power of a power rule: $2^{3 \times 4} = 2^{12}$.

68. What is the value of $(3^2)^4$?

    $3^8$. Power of a power: $(3^2)^4 = 3^{2 \cdot 4} = 3^8$.

69. State the power of a power rule using $(a^m)^n$.

    $(a^m)^n = a^{mn}$. When raising a power to another power, multiply the exponents.

70. Identify the simplified form of $\frac{2^3 \cdot 2^{-5}}{2^{-1}}$ as a single power of $2$.

    $2^{-1}$. $\frac{2^{3+(-5)}}{2^{-1}} = \frac{2^{-2}}{2^{-1}} = 2^{-2-(-1)} = 2^{-1}$.

71. What is the value of $\frac{5^9}{5^2}$?

    $5^7$. Quotient rule: $\frac{5^9}{5^2} = 5^{9-2} = 5^7$.

72. State the power of a power rule for $(a^m)^n$.

    $(a^m)^n = a^{mn}$. When raising a power to a power, multiply the exponents.

73. What is $(4x^2)^3$ written using exponents? (Assume $x$ is any real number.)

    $64x^6$. Apply power of a product: $4^3 \cdot (x^2)^3 = 64x^6$.

74. What is the power of a power rule: $(a^m)^n =$ ?

    $a^{mn}$. When raising a power to a power, multiply the exponents.

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