Math Quiz 5 Page 5 Name ✓ Solved

Math 012quiz 5page 5math 012 Quiz 5name

Simplify the given expressions, show all work, and verify your solutions by raising your simplified answer to the relevant power as specified. Complete the quiz individually, including the required declaration statement at the end, and submit it by the due date. Use textbooks, notes, and online resources but do not seek external help.

Sample Paper For Above instruction

The purpose of this paper is to demonstrate the process of simplifying complex algebraic expressions and verifying the solutions through exponentiation, consistent with the instructions provided for the Math 012 Quiz 5. The methodology involves step-by-step calculations, rationalization where necessary, and proper notation, emphasizing clarity and precision throughout.

Introduction

Mathematics, especially algebra, requires not only finding solutions but also verifying their correctness through inverse operations. The process of simplifying expressions and confirming the results ensures comprehension and accuracy, fundamental skills in mathematical problem-solving. This paper details the approach to solving ten algebraic problems, including exponentiation, rationalization, and radical notation, according to the specified instructions.

Problem 1: Simplify and Verify by Raising to the 2nd Power

Suppose the expression to simplify is \( \sqrt{196} \). The initial step involves identifying the perfect square factors. Since 196 is a perfect square, its square root is \( \sqrt{196} = 14 \). To verify, we raise the simplified answer back to the 2nd power: \( 14^2 = 196 \), confirming the correctness.

Similarly, if the problem involves more complex radicals or algebraic expressions, the process includes simplifying inside the radical, applying properties of exponents, and confirming the result through inverse operations.

Problem 2: Simplify and Verify by Raising to the 3rd Power

For example, simplifying \( \sqrt[3]{54} \) involves factoring 54 into prime factors: \( 54 = 2 \times 3^3 \). The cube root then simplifies to \( \sqrt[3]{2 \times 3^3} = 3 \times \sqrt[3]{2} \). To verify, raise this to the third power: \( (3 \times \sqrt[3]{2})^3 = 3^3 \times (\sqrt[3]{2})^3 = 27 \times 2 = 54 \), confirming accuracy.

Problem 3: Simplify and Verify by Raising to the 4th Power

The same logic applies when simplifying \(\sqrt[4]{x^8}\). Recognizing that \(\sqrt[4]{x^8} = x^{8/4} = x^{2}\), then raising to the 4th power yields \((x^2)^4 = x^{8}\), which matches the original expression's power, validating the simplification.

Problem 4-6: Perform and Simplify Operations

These questions involve combining operations such as addition, subtraction, multiplication, and division of algebraic terms, sometimes requiring rationalization of denominators. For example, rationalizing the denominator involves multiplying numerator and denominator by the conjugate of the denominator, e.g., for \(\frac{1}{\sqrt{2}}\), multiply numerator and denominator by \(\sqrt{2}\) to obtain \(\frac{\sqrt{2}}{2}\).

Problem 7-9: Rationalize Denominators

This includes specific strategies such as multiplying numerator and denominator by the conjugate when dealing with binomials involving radicals, or simplifying complex fractions. For instance, \(\frac{1}{\sqrt{3} + 2}\) becomes \(\frac{\sqrt{3} - 2}{(\sqrt{3} + 2)(\sqrt{3} - 2)} = \frac{\sqrt{3} - 2}{3 - 4} = \frac{\sqrt{3} - 2}{-1} = 2 - \sqrt{3}\).

Problem 10: Rewrite Expression Using Radical Notation

Expressing algebraic expressions in radical notation involves converting fractional exponents into roots. For example, \(x^{1/3}\) is rewritten as \(\sqrt[3]{x}\). Simplifications may include combining radicals, factoring under radicals, or expressing sums/differences of radicals in simpler forms.

Conclusion

Effective algebraic manipulation hinges on understanding properties of exponents, radicals, and rationalization techniques. Verifying solutions via inverse operations fortifies the correctness of the work and enhances conceptual understanding. Adhering to methodical steps and meticulous notation ensures clarity and accuracy, essential in mastering algebraic simplifications.

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