W2 College Algebra I84 Raising A Product Or A Quotient To A

W2 College Algebra I84 Raising A Product Or A Quotient To A Power

Perform a series of algebraic simplifications and operations involving exponents, radicals, and complex numbers as specified in the assignment. The tasks include simplifying expressions with powers, roots, and absolute values, identifying bases and exponents, converting numbers between decimal and scientific notation, evaluating functions, and working with complex numbers in standard form. Assume all denominators are non-zero, and that 0^0 is not considered, unless explicitly specified. Simplify all expressions using positive exponents where possible and express the final answers in exact form, including radicals and fractions as needed.

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Algebraic operations involving powers, roots, and complexes are fundamental skills in college algebra, enabling students to manipulate and simplify complex expressions systematically. The goal of this assignment is to develop proficiency in applying exponent rules, radical simplifications, and complex number arithmetic in various contexts.

Simplification of Expressions with Powers and Roots

One of the core tasks involves raising products or quotients to powers. For example, simplifying (x^5 / -3y^3)^4 requires applying the power rule for exponents: each component inside the parentheses is raised to the power. This rule states that (a / b)^n = a^n / b^n, and similarly, (a b)^n = a^n b^n. Applying this, we get x^{20} / (-3)^4 y^{12}. Since (-3)^4 = 81, the expression simplifies to x^{20} / (81 y^{12}).

Similarly, simplifying expressions like (a^4b^8)(a^6b^7) involves the product rule for exponents: a^{m} a^{n} = a^{m + n}, and b^{p} b^{q} = b^{p + q}. Therefore, this simplifies to a^{4+6}b^{8+7} = a^{10}b^{15}.

Division of like bases follows the quotient rule for exponents: a^{m} / a^{n} = a^{m - n}, provided a ≠ 0. For example, 4b^{8} / 2b^{4} simplifies to (4/2) b^{8-4} = 2b^{4}. When raising expressions to powers, the rule (a^{m})^{n} = a^{m*n} applies, as in (x^{6})^{4} = x^{24}.

Roots are expressed as fractional exponents, where √(x) = x^{1/2}. For example, √(16t) = (16t)^{1/2} = 4 t^{1/2}. If a root cannot be simplified further, such as √(7), it should be stated explicitly.

Complex expressions involving negative exponents are handled through the rule a^{-n} = 1 / a^n. For instance, R^{-7} can be expressed as 1 / R^{7} for positive exponents.

Expressing numbers like 2.68×10^{-5} in decimal notation involves moving the decimal point five places to the left, resulting in 0.0000268. Conversely, numbers like 6,500 are expressed in scientific notation as 6.5 × 10^{3}.

Evaluation of functions such as F(t) = √(t^2 + 1), involves substituting the given value for t and simplifying. For example, F(3) = √(3^2 + 1) = √10.

Radicals can be simplified if the radicand has perfect square factors. For instance, √(-8c)^2 simplifies to | -8c | = 8 |c|, because the absolute value ensures a nonnegative result.

Expressions involving irrational or imaginary terms, like √(-1), involve the imaginary unit i, where √(-1) = i. Complex number operations such as subtraction (7 – i) – (9 + 5i) and multiplication -7i * 2i are performed using distributive properties and combining like terms, with results expressed in a + bi form.

For negative radicands, the imaginary unit is employed to maintain real or complex representations, as in √(-49) = 7i, since √(-49) = √(49) * √(-1) = 7i.

In summary, mastering these algebraic manipulations prepares students to handle more advanced mathematical problems, including functions, complex numbers, and their applications in various scientific fields. The key is applying the correct exponent rules, radical simplifications, and handling complex numbers rigorously.

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