Problem 50 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ \frac{20 x^{24}}{10 x^{6}} $$

Step-by-Step Solution

Answer

The simplification of the expression is \(2x^{18}\).

Step 1: Simplifying the Numerical Coefficients

The expression has numerical coefficients \(20\) and \(10\) which are divided. This gives the simplified coefficient as \(20/10 = 2\).

Step 2: Applying the Rules of Exponents

Next, we want to simplify \(x^{24}/x^{6}\). For exponents, when we have division, we subtract the exponents. Therefore it becomes \(x^{(24-6)}\).

Step 3: Simplify the Exponent Terms

After subtracting, we will have \(x^{18}\).

Key Concepts

Rules of Exponents

Understanding how exponents work is essential for simplifying exponential expressions. When dealing with exponents, there are several rules one must keep in mind, notably the exponent 'laws'. For example, when you multiply powers that have the same base, you add the exponents:
\( x^a \times x^b = x^{(a+b)} \). Conversely, when dividing powers with the same base, you subtract the exponents:
\( \frac{x^a}{x^b} = x^{(a-b)} \). This latter rule is extremely useful in simplification, as seen in our exercise example where the terms \( x^{24} \) and \( x^6 \) are divided. Write these rules down—they will come in handy often in algebraic calculations.

Numerical Coefficients

In algebra, numerical coefficients are the numbers that multiply the variables or letters. Simplifying these first makes the rest of the problem far more approachable. To simplify an expression with numerical coefficients, like \(\frac{20}{10}\) from our problem, you just perform basic division or multiplication, as required. Here, it simplifies to 2. This same principle applies whether the numbers are whole, fractions, or decimals. Treat the numerical coefficients separately from the variable terms to streamline the simplification process.

Algebraic Simplification

Algebraic simplification is about making complex expressions easier to handle. This typically involves combining like terms, reducing fractions, or using the distributive property to eliminate parentheses. A key aspect of simplification is to perform operations systematically: start with coefficients, as you already learned, and then apply the rules of exponents. By taking this step-by-step approach, even the most daunting algebraic expression can be broken down into something more manageable.

Exponent Subtraction

When an algebraic expression involves division of like bases, you need to subtract the exponents. This rule is a part of the fundamental rules of exponents and is crucial for simplification. In our textbook exercise, subtracting the exponent of the denominator from the exponent of the numerator \(24 - 6\) gives us an exponent of 18 for the variable \(x\). It's important to carefully subtract the smaller exponent from the larger to ensure accuracy in your final answer. Remember, exponent subtraction only applies when you are dividing terms with the same base; otherwise, different rules are used.