Problem 10 - 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: Simplify (\(\sqrt{x})^{2}\)

The expression \((\sqrt{x})^{2}\) involves squaring a square root. Applying the property \((\sqrt{a})^{2} = a\), we can simplify the expression to \(x\).

Step 2: Simplify (\(\sqrt{x-5})^{2}\)

The expression \((\sqrt{x-5})^{2}\) is also a square root raised to the power of 2. Using the same property as Step 1, \((\sqrt{a})^{2} = a\), the expression simplifies to \(x-5\).

Step 3: Simplify (\(\sqrt[3]{4x-8})^{3}\)

Here, \((\sqrt[3]{4x-8})^{3}\) involves raising a cube root to the third power. Using the property \((\sqrt[3]{a})^{3} = a\), it simplifies to \(4x-8\).

Step 4: Simplify (\(\sqrt[4]{8x})^{4}\)

The expression \((\sqrt[4]{8x})^{4}\) involves raising a fourth root to the power of 4. By applying \((\sqrt[n]{a})^{n} = a\), the expression becomes \(8x\).

Step 5: Simplify (\(4 \sqrt{2x})^{2}\)

First, identify that this is a product inside a square. Rewrite \(4\sqrt{2x}\) as \((4) \cdot (\sqrt{2x})\) and square each part using \((ab)^2 = a^2 b^2\). This results in \(4^2 \cdot (\sqrt{2x})^2 = 16 \cdot 2x = 32x\).

Step 6: Simplify (\(3 \sqrt[3]{x+1})^{3}\)

Break down \(3 \sqrt[3]{x+1}\) into \((3) \cdot (\sqrt[3]{x+1})\) and apply the cube root exponent property: \((ab)^3 = a^3 b^3\). Cube each part: \(3^3 (\sqrt[3]{x+1})^3 = 27(x+1)\). Finally, simplify to \(27x + 27\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent or power. It can be thought of as repeated multiplication of the base number. For example, when we say \(3^4\), we mean 3 multiplied by itself 4 times, which is \(3 \times 3 \times 3 \times 3\). Exponentiation is often used to simplify expressions and solve equations with small or large numbers.

There are several properties of exponents that make calculations easier and allow for simplification:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{mn}\)
• Zero Exponent: Any number raised to the power of zero is 1, or \(a^0 = 1\), provided that \(aeq0\).
• Negative Exponent: Indicates the reciprocal effect: \(a^{-n} = \frac{1}{a^n}\)

These properties are essential for simplifying exponential expressions, as seen in our examples where square and cube roots are raised to powers.

Radicals

Radicals refer to expressions involving roots, such as square roots, cube roots, and so on. The radical symbol \(\sqrt{}\) represents the square root, whereas different roots are indicated by a number above the radical sign, such as the cube root \(\sqrt[3]{}\).

A key property of radicals that is useful in simplifying them is \((\sqrt[n]{a})^n = a\). This means raising a radical to the power that equals its root results in the radicand (the number under the root sign). For instance, in our example \((\sqrt{x})^2 = x\), squaring the square root of \(x\) simply returns \(x\).

Radicals can be simplified in expressions through such properties, making them a vital zone of algebraic manipulation. When combined with powers, they enable us to transform and interface expressions in key algebraic solutions.

Properties of Exponents

The properties of exponents help ease complex calculations and clarify expressions involving powers. Understanding these properties ensures that expressions can be simplified effectively, especially when dealing with algebraic terms. Let’s summarize the primary properties:

• Product Rule: \(a^m \cdot a^n = a^{m+n}\) - When multiplying two powers with the same base, keep the base and add the exponents.
• Power Rule: \((a^m)^n = a^{mn}\) - When raising a power to another power, multiply the exponents.
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\) - When dividing two powers with the same base, keep the base and subtract the exponents.
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\) - A negative exponent shows the reciprocal of the base raised to the positive exponent.

Each of these properties allows us to tackle various calculations by transforming complex multiplications and divisions into more manageable forms.

Algebraic Expressions

Algebraic expressions consist of variables, coefficients, and constants combined using operations like addition, subtraction, multiplication, and division. These expressions form the backbone of algebra and empower us to solve real-world problems.

When simplifying expressions, recognizing and manipulating their components is key. For example:

• Terms: Parts of an algebraic expression separated by addition or subtraction.
• Factors: Numbers or expressions that are multiplied together to get another number or expression.
• Simplification: Involves using arithmetic and algebraic properties to transform expressions into their simplest form, such as combining like terms or using properties of exponents.

By deeply understanding algebraic expressions, you can simplify complex problems and enable clearer, concise solutions. This is showcased when simplifying expressions with squares and radicals, breaking them down into understandable components.