Problem 10 - Elementary and Intermediate Algebra

Question

Simplify each expression. a. \((\sqrt{x})^{2}\) b. \((\sqrt{x-5})^{2}\) c. \((\sqrt[3]{4 x-8})^{3}\) d. \((\sqrt[4]{8 x})^{4}\) e. \((4 \sqrt{2 x})^{2}\) f. \((3 \sqrt[3]{x+1})^{3}\)

Step-by-Step Solution

Answer

a. \(x\), b. \(x-5\), c. \(4x-8\), d. \(8x\), e. \(32x\), f. \(27x+27\).

Step 1: Examine the First Expression

Look at the expression \((\sqrt{x})^{2}\). The square of the square root of any number is the number itself. Thus, \((\sqrt{x})^{2} = x\).

Step 2: Simplify the Second Expression

For \((\sqrt{x-5})^{2}\), the square of the square root cancels out, so \((\sqrt{x-5})^{2} = x-5\).

Step 3: Address the Third Expression

Consider \((\sqrt[3]{4x-8})^{3}\). The cube of the cube root of any number is the number itself. Therefore, \((\sqrt[3]{4x-8})^{3} = 4x - 8\).

Step 4: Simplify the Fourth Expression

Analyze \((\sqrt[4]{8x})^{4}\). The fourth power of the fourth root of a number returns the number itself. Thus, \((\sqrt[4]{8x})^{4} = 8x\).

Step 5: Compute the Fifth Expression

Examine \((4\sqrt{2x})^{2}\). First, recognize that this is the square of a product: \((4\sqrt{2x})^{2} = 4^{2} \times (\sqrt{2x})^{2} = 16 \times 2x = 32x\).

Step 6: Simplify the Sixth Expression

Consider \((3\sqrt[3]{x+1})^{3}\). Since this is the cube of a product: \((3\sqrt[3]{x+1})^{3} = 3^{3} \times (\sqrt[3]{x+1})^{3} = 27 \times (x+1) = 27x + 27\).

Key Concepts

Square Roots and Their Properties

Square roots are a fundamental concept in algebra, symbolized by the radical sign \(\sqrt{}\). A square root of a number \(a\) is a value that, when multiplied by itself, gives \(a\). For example, the square root of 9 is 3, because \(3 \times 3 = 9\). Understanding square roots is crucial for simplifying expressions.

One key property of square roots is that the square of the square root of a number results in the original number. Mathematically, this is expressed as \((\sqrt{x})^2 = x\). This property is commonly used to simplify expressions where square roots are raised to powers, like in the originals: \((\sqrt{x-5})^2 = x-5\). This simplification occurs because the operations of taking a square root and squaring are inverse operations, effectively canceling each other out.

• When simplifying square root expressions, look for opportunities to apply the inverse relationship between squaring and taking the square root.
• Always ensure the expression under the square root is non-negative, as square roots of negative numbers lead to imaginary results.

Cube Roots and Their Applications

Cube roots follow a similar principle to square roots but involve exponents of three. The cube root of a number \(a\), denoted as \(\sqrt[3]{a}\), is the number which, when used in a multiplication three times, equals \(a\). For example, since \(3 \times 3 \times 3 = 27\), \(\sqrt[3]{27} = 3\).

Cube roots have special significance when simplifying expressions like \((\sqrt[3]{4x-8})^3 = 4x-8\). Here, the cube and the cube root cancel each other, due to being inverse operations.

• Cube roots are particularly used when dealing with volumes and scaling across all three dimensions.
• Unlike square roots, cube roots can be applied to negative numbers, resulting in negative answers (e.g., \(\sqrt[3]{-8} = -2\)).

Understanding cube roots can significantly simplify algebraic expressions by recognizing the patterns of cancellation between cube powers and cube roots.

Exponents and Their Role

Exponents serve as a shorthand for repeated multiplication of a number by itself. In expressions like the fifth and sixth from the original problem, exponents play a crucial role. For example, in \((4\sqrt{2x})^2 = 32x\), the exponent 2 indicates we're multiplying \(4\sqrt{2x}\) by itself.

Important properties of exponents include:

• Power of a Power: \((a^m)^n = a^{m\times n}\), useful in expressions with multiple nested exponents.
• Distribution: When distributing exponents within products, \((ab)^n = a^n \times b^n\).

Using these properties, simplifying expressions becomes more manageable, especially when expressions contain both roots and powers. Recognizing and applying these exponent rules allows mathematicians to rearrange and simplify complex expressions easily.

Understanding Polynomial Expressions

Polynomial expressions consist of variables and constant coefficients combined using the operations of addition, subtraction, multiplication, and non-negative integer exponents. The original expression \((3\sqrt[3]{x+1})^3 = 27x + 27\) simplifies into a polynomial form by understanding the behavior of cube roots and exponents.

Polynomials have certain characteristics:

• They are classified by degree, which is the highest power of the variable in the expression.
• Simplification involves combining like terms and using the laws of exponents.

In applications of algebra, polynomial expressions allow for easy evaluation of equations, prediction of function behavior, and solving for unknowns. By understanding how to manipulate and simplify polynomials, you lay a foundation for more advanced mathematical operations, such as factoring and finding roots.