Math 9 Review Assignment Notes

Exponents Review

Math 9 Review Assignment Worksheet Page 1

Exponent Form, Expanded Form, and Standard Form Table

  • Row 1:

    • Exponent Form: 454^5
    • Expanded Form: 4×4×4×4×44 \times 4 \times 4 \times 4 \times 4
    • Standard Form: 10241024

  • Row 2:

    • Exponent Form: 353^5
    • Expanded Form: 3×3×3×3×33 \times 3 \times 3 \times 3 \times 3
    • Standard Form: 243243

  • Row 3:

    • Exponent Form: 262^6
    • Expanded Form: 2×2×2×2×2×22 \times 2 \times 2 \times 2 \times 2 \times 2
    • Standard Form: 6464

  • Row 4:

    • Exponent Form: (−2)5(-2)^5
    • Expanded Form: (−2)×(−2)×(−2)×(−2)×(−2)(-2) \times (-2) \times (-2) \times (-2) \times (-2)
    • Standard Form: −32-32

Simplifying Exponential Expressions Without a Calculator

  • Fundamental Rules of Exponents:

    • Product Rule: am×an=am+na^m \times a^n = a^{m+n}
    • Quotient Rule: am÷an=am−na^m \div a^n = a^{m-n}
    • Power of a Power Rule: (am)n=am×n(a^m)^n = a^{m \times n}
    • Zero Exponent Rule: a0=1a^0 = 1 for any non-zero base aa
    • Power of a Quotient Rule: (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

  • Solutions and Step-by-Step Simplification:

    • a) 145×14314^5 \times 14^3

    • Apply the Product Rule: 145+314^{5+3}

    • Result: 14814^8

    • b) 611÷666^{11} \div 6^6

    • Apply the Quotient Rule: 611−66^{11-6}

    • Result: 656^5

    • c) (32)5(3^2)^5

    • Apply the Power of a Power Rule: 32×53^{2 \times 5}

    • Result: 3103^{10}

    • d) 810÷88^{10} \div 8

    • Rewrite base exponent as 818^1: 810÷818^{10} \div 8^1

    • Apply the Quotient Rule: 810−18^{10-1}

    • Result: 898^9

    • e) 12012^0

    • Apply the Zero Exponent Rule

    • Result: 11

    • f) (23)3\left(\frac{2}{3}\right)^3

    • Apply the Power of a Quotient Rule

    • Result: 2333\frac{2^3}{3^3}

Polynomials Review

Classification of Polynomials Table

  • Row 1:

    • Expression: 4m2+74m^2 + 7
    • Degree: 22
    • Number of Terms: 22
    • Name: Binomial

  • Row 2:

    • Expression: 5m2n45m^2 n^4
    • Degree: 2+4=62 + 4 = 6
    • Number of Terms: 11
    • Name: Monomial

  • Row 3:

    • Expression: 2b5+4b3−12b^5 + 4b^3 - 1
    • Degree: 55
    • Number of Terms: 33
    • Name: Trinomial

Combining Like Terms

  • a) −11m+6m-11m + 6m

    • Group numerical coefficients: (−11+6)m(-11 + 6)m
    • Result: −5m-5m

  • b) 8b−12b−3b8b - 12b - 3b

    • Combine first two terms: 8b−12b=−4b8b - 12b = -4b
    • Combine remaining term: −4b−3b-4b - 3b
    • Result: −7b-7b

  • c) −4b3+4b−3b3-4b^3 + 4b - 3b^3

    • Group like terms with variable factor b3b^3: (−4b3−3b3)+4b(-4b^3 - 3b^3) + 4b
    • Result: −7b3+4b-7b^3 + 4b

  • d) 7x2m+11m−6x2m7x^2 m + 11m - 6x^2 m

    • Group like terms with variable factor x2mx^2 m: (7x2m−6x2m)+11m(7x^2 m - 6x^2 m) + 11m
    • Simplify coefficient 7−6=17 - 6 = 1: 1x2m+11m1x^2 m + 11m
    • Result: x2m+11mx^2 m + 11m

Adding and Subtracting Polynomials

  • a) (−11a−12v)+(15a+6v)(-11a - 12v) + (15a + 6v)

    • Remove parentheses: −11a−12v+15a+6v-11a - 12v + 15a + 6v
    • Rearrange like terms: (−11a+15a)+(−12v+6v)(-11a + 15a) + (-12v + 6v)
    • Result: 4a−6v4a - 6v

  • b) (3b−8b2+4)+(−12b2−4b+1)(3b - 8b^2 + 4) + (-12b^2 - 4b + 1)

    • Remove parentheses: 3b−8b2+4−12b2−4b+13b - 8b^2 + 4 - 12b^2 - 4b + 1
    • Rearrange in descending degree order: (−8b2−12b2)+(3b−4b)+(4+1)(-8b^2 - 12b^2) + (3b - 4b) + (4 + 1)
    • Result: −20b2−b+5-20b^2 - b + 5

  • c) (4b−11)−(8b+9)(4b - 11) - (8b + 9)

    • Distribute the negative sign across the second expression: 4b−11−8b−94b - 11 - 8b - 9
    • Group like terms: (4b−8b)+(−11−9)(4b - 8b) + (-11 - 9)
    • Result: −4b−20-4b - 20

  • d) (3b−8b2+4)−(−4b2+3b−1)(3b - 8b^2 + 4) - (-4b^2 + 3b - 1)

    • Distribute the negative sign across the second expression: 3b−8b2+4+4b2−3b+13b - 8b^2 + 4 + 4b^2 - 3b + 1
    • Group like terms: (−8b2+4b2)+(3b−3b)+(4+1)(-8b^2 + 4b^2) + (3b - 3b) + (4 + 1)
    • Note that 3b−3b=03b - 3b = 0
    • Result: −4b2+5-4b^2 + 5