7-1: Integer Exponents
38. For the expression ((\frac{2}{3v})^{-3}) when (v = 9): First, substitute (v) with 9: ((\frac{2}{3 \times 9})^{-3}) Simplify the fraction: ((\frac{2}{27})^{-3}) Now, to deal with the negative exponent, take the reciprocal and change the exponent to positive: ((\frac{27}{2})^{3}) Finally, calculate the cube: ((\frac{27}{2})^{3} = \frac{19683}{8})
45. For the expression ((2x2)2 \times (3x3)3): First, apply the power of a power rule: (4x^4 \times 27x^9) Then, multiply the coefficients and add the exponents: (108x^{13})
49. For the expression (\frac{2f0}{7g{-10}}): Anything raised to the power of 0 is 1, so (f^0 = 1). For (g^{-10}), take the reciprocal to make the exponent positive: (g^{10}). So, the expression simplifies to: (\frac{2}{7g^{10}})
50. For the expression (\frac{r{-5}}{s{-1}}): Take the reciprocal of both (r^{-5}) and (s^{-1}) to make the exponents positive: (r^5) and (s). So, the expression simplifies to: (r^5s)
7-3: Multiplication Properties of Exponents
47. For the expression ((5x2)(5x2)^2): First, apply the power to the second term: (5x^2 \times 25x^4) Then, multiply the coefficients and add the exponents: (125x^6) Note: The exponent 2 applies only to (x), not to 5.
7-4: Division Properties of Exponents
10. For the expression ((\frac{x2}{xy3})^3): First, apply the exponent to both numerator and denominator: (\frac{x6}{x3y^9}) Then, simplify by subtracting the exponents in the numerator and denominator: (x{6-3}y{-9}) So, the expression simplifies to: (x3y{-9}) or (\frac{x3}{y9})
11. For the expression ((\frac{a3}{(a3b)2})2): First, apply the power to the denominator: (\frac{a3}{a6b^2}) Then, apply the power to the entire fraction: (\frac{a6}{a{12}b^4}) Finally, simplify by subtracting the exponents: (a{6-12}b{-4}) So, the expression simplifies to: (a{-6}b{-4}) or (\frac{1}{a6b4})
7-6: Polynomials
9. For the expression (075a^2b - 2a3b5): This seems to be a typo. If it’s meant to be (0.75a^2b - 2a3b5), then there’s nothing to simplify unless there’s more context or additional terms.
14. For the expression (-2b + 5 + b^2): This is already simplified. It’s a polynomial in standard form.
16. For the expression (5s^2 - 3s + 3 - s^7): This is also already simplified. It’s a polynomial in standard form.
17. For the expression (2x + 3x^2 - 1): This is already simplified. However, if you want it in standard form, it should be written as (3x^2 + 2x - 1).
