Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(\frac{x \cdot x^{1 / 2}}{x^{3 / 2}}\)

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Break down the equation

The expression is \(\frac{x \cdot x^{1 / 2}}{x^{3 / 2}}\). Remember that when you multiply with the same bases, you add the exponents. We can rewrite the numerator as \(x^1 \cdot x^{1 / 2}\), which simplifies to \(x^{(1 + 1 / 2)}\). The denominator remains the same which is \(x^{3 / 2}\).

Step 2: Simplify the numerator

The numerator \(x^{(1+1/2)}\) simplifies to \(x^{3/2}\). Hence the expression now reads \(\frac{x^{3 / 2}}{x^{3 / 2}}\).

Step 3: Simplify the fraction

When you divide with the same base, you subtract the exponents. Here both numerator and denominator have the same base and exponent so the result is 1. The subtracted exponent becomes \(3/2 - 3/2 = 0\), and any number (except for 0) raised to the 0th power equals 1.

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It involves reducing expressions to their simplest form by performing arithmetic operations and applying mathematical properties. A simplified expression is usually shorter and easier to understand or evaluate.
To simplify expressions, it is important to:

• Identify common factors or like terms that can be combined.
• Apply arithmetic operations such as addition, subtraction, multiplication, and division.
• Use the properties of exponents and other mathematical rules.

In our exercise, the expression \(\frac{x \cdot x^{1/2}}{x^{3/2}}\) was simplified using the properties of exponents. By simplifying, we not only make expressions easier to work with but also gain deeper insight into their structure.

Properties of Exponents

The properties of exponents are essential rules in algebra that help efficiently solve expressions involving powers. Understanding these rules allows us to manipulate expressions consistently and logically. There are several crucial properties:

• Product of Powers: When multiplying two exponential terms with the same base, add the exponents: \(x^a \cdot x^b = x^{a+b}\)
• Quotient of Powers: When dividing two exponential terms with the same base, subtract the exponents: \(x^a / x^b = x^{a-b}\)
• Power of a Power: When raising a power to another power, multiply the exponents: \((x^a)^b = x^{a\cdot b}\)
• Zero Exponent: Any base raised to the zero power equals one: \(x^0 = 1\), provided that \(x eq 0\)

In the step-by-step solution, these properties were applied to combine and simplify exponents, particularly in transforming the expression and its eventual simplification into one.

Rational Exponents

Rational exponents extend the concept of exponents to fractional powers, providing a notation that is often more convenient than radical symbols, particularly for complex calculations. A rational exponent \(x^{m/n}\) is another way to express \(\sqrt[n]{x^m}\). For example, \(x^{1/2}\) represents the square root of \(x\), \(x^{1/3}\) represents the cube root, and so on.
Here are some tips for dealing with rational exponents:

• Convert complex expressions involving roots into rational exponent notation for ease of manipulation.
• Apply the same exponent rules to rational exponents, such as multiplication and division of like bases.
• Rational exponents allow us to seamlessly integrate roots into algebra, making equations and expressions more versatile.

In the exercise above, rational exponents were used to simplify the expression by breaking down and reassembling the fractional powers, ultimately making calculation smoother and more intuitive.