Express Your Answer With Positive Exponents 1 3xy 14 D

Simply Express Your Answer With Positive Exponents1 3xy 14 Div

Simplify the expressions below, ensuring all exponents are positive. Perform any necessary algebraic manipulations, including factoring, expanding, or combining like terms, and express the final answer with only positive exponents.

Evaluate or simplify the following numerical expressions and algebraic expressions, providing exact or simplified forms as appropriate:

  1. (3xy-1)4 ÷ z2
  2. (3xy2)2 ÷ x3
  3. (–a2b)3 ÷ 2b-1
  4. (4x)3 ÷ y3

Evaluate or simplify these numerical expressions:

  1. (16)3/4
  2. (4)3/2

Express with positive exponents:

  1. 62 × 6-2
  2. ₁₁1/2 × 113

Solve the following quadratic equations using the quadratic formula and simplify the answers:

  1. x2 - 3x – 1 = 0
  2. 4x2 + x – 1 = 0
  3. 6x2 - 2x = 0

Combine and simplify the radicals:

  1. 3√13 + 7√13
  2. √90 – √10
  3. √75a3 + √12a

Paper For Above instruction

In algebra, simplifying expressions with positive exponents is fundamental to maintaining clarity and coherence in mathematical operations. The goal is to manipulate the given expressions so that all exponents are positive, which often involves utilizing properties of exponents such as the quotient rule, product rule, and power rule. These properties facilitate rewriting expressions to a form that is more straightforward and easier to interpret, especially in complex algebraic and numeric calculations.

Let us analyze and simplify each provided expression step by step, adhering to the principle of using only positive exponents. Starting with the algebraic expressions:

Algebraic Simplifications

  1. (3xy-1)4 ÷ z2
  2. Applying the power rule: (ab)n = anbn, we get 34 x4 y-4. To rewrite with positive exponents, y-4 becomes 1/y4. Dividing by z2 is equivalent to multiplying by z-2, which we convert to 1/z2. Assembling all parts, the simplified expression is:
  • 34 x4 / y4 z2 = 81 x4 / y4 z2
  • (3xy2)2 ÷ x3

    • Apply the power rule: 32 x2 y4. Since dividing by x3 is equivalent to subtracting exponents, we have:

    • 9 x2 y4 / x3 = 9 y4 x-1

    To express with only positive exponents, write x-1 as 1/x, giving:

    • 9 y4 / x
  • (–a2b)3 ÷ 2b-1

    • Expanding the numerator: (–1)3 a6 b3 = –a6 b3. Dividing by 2b-1 is like multiplying by 1/(2b-1) = 1/2 * b1. Combining these, the expression becomes:

    • (–a6 b3) / 2b-1 = (–a6 b3) * (b / 2) = –a6 b4 / 2
  • (4x)3 ÷ y3

    • Applying the power rule: 43 x3 ÷ y3 = 64 x3 / y3

    Numerical Expressions

    1. (16)3/4
    2. Express 16 as 24; then (24)3/4 = 24 * 3/4 = 23 = 8
    3. (4)3/2
    4. Express 4 as 22; then (22)3/2 = 22 * 3/2 = 23 = 8

    Exponents and Powers

    1. 62 × 6-2
    2. This is equivalent to 62 / 62 = 36 / 62 = 36 / 36 = 1
    3. ₁₁1/2 × 113
    4. ₁₁1/2 indicates the square root of 11, which is √11. Thus, the expression becomes √11 × 113. In decimal form, √11 ≈ 3.317; multiply: 3.317 × 113 ≈ 375.22. For exact form, it remains as √11 × 113.

    Quadratic Equations

    To solve quadratic equations, the quadratic formula is applicable: x = [–b ± √(b2 – 4ac)] / 2a. Let us apply this to the given equations:

    Equation 8: x2 – 3x – 1 = 0

    Identify coefficients: a=1, b=–3, c=–1

    Calculate discriminant: D = (–3)2 – 4 × 1 × (–1) = 9 + 4 = 13

    Solutions: x = [3 ± √13] / 2

    Equation 9: 4x2 + x – 1 = 0

    Coefficients: a=4, b=1, c=–1

    Discriminant: D = 12 – 4 × 4 × (–1) = 1 + 16 = 17

    Solutions: x = [–1 ± √17] / 8

    Equation 10: 6x2 – 2x = 0

    Rewrite as: 6x2 – 2x = 0, or x(6x – 2) = 0

    Solutions are x=0 or x= (2/6) = 1/3

    Radical Expression Simplifications

    1. 3√13 + 7√13
    2. Combine like radicals: (3 + 7)√13 = 10√13
    3. √90 – √10
    4. Simplify each radical: √90 = √(9×10) = 3√10; thus, 3√10 – √10 = (3 – 1)√10 = 2√10
    5. √75a3 + √12a
    6. Simplify √75a3 = √(25×3×a3) = 5√3 a1.5. Rewrite a1.5 as a × √a.
    7. Similarly, √12a = √(4×3×a) = 2√3 a0.5
    8. Thus, the combined expression becomes: 5√3 a × √a + 2√3 a0.5. Recognizing common factors, the expression can be written as √3 a (5 a + 2), simplifying the radical expressions further when necessary.

    Conclusion

    By systematically applying the properties of exponents and radicals — including the quotient rule, product rule, and power rule — we are able to rewrite all algebraic and numeric expressions with only positive exponents. Solving quadratic equations with the quadratic formula involves calculating the discriminant and simplifying square roots, leading to exact solutions. The radicals are simplified by factoring out perfect squares, leading to expressions in simplest radical form. Mastery of these algebraic manipulations is essential for higher-level mathematics, providing clarity and ensuring consistency across various types of mathematical expressions.

    References

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    • Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
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