Problem 35 - College Algebra

Question

Simplify each exponential expression $$ \frac{x^{14}}{x^{7}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{x^{14}}{x^{7}}\) is \(x^7\).

Step 1: Identify the Problem

We have an exponential expression of the form \(x^n/x^m\), where \(n=14\) and \(m=7\). We need to simplify this expression. The property of exponents we are using is \(x^n/x^m = x^{(n-m)}\).

Step 2: Subtract the Exponents

Subtract the exponent of the denominator from the exponent of the numerator. This gives us \(x^{(14-7)}\).

Step 3: Simplify the Exponents

Perform the subtraction to find the new exponent. This results in \(x^7\) as our simplified expression.

Key Concepts

Laws of Exponents

Exponential expressions can seem tricky at first, but the laws of exponents make them much easier to manage. One of the essential laws that helps simplify such expressions is the Quotient of Powers Property. This property tells us that when we divide like bases, we can subtract the exponents: \(x^n/x^m = x^{(n-m)}\). This principle emerges from the idea that exponents indicate repeated multiplication.

For example, \(x^n\) means multiplying the base \(x\), \(n\) times. Similarly, \(x^m\) means multiplying it \(m\) times. When you divide these, the repeated \(x\)'s cancel out.

Important points to remember about the laws of exponents include:

• Multiply Powers: Add the exponents \(x^a \times x^b = x^{(a+b)}\).
• Power of a Power: Multiply the exponents \((x^a)^b = x^{(a\times b)}\).
• Zero Exponent Rule: Any base with a zero exponent equals one \(x^0 = 1\), given that \(x eq 0\).

These rules simplify working with exponentials significantly, letting you handle even complex expressions with confidence.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It involves reducing an expression to its simplest form, making it easier to understand or solve. In the case of exponents, this often means using properties to condense the expression.

In our problem, we began with \(x^{14}/x^7\). By applying the quotient property of exponents, we subtracted the denominator's exponent from the numerator's. This gave us \(x^{14 - 7}\). Simplifying this small equation, we reach \(x^7\), which is the simplest form of the expression.

• Identify Common Bases: Focus on terms with the same base to see how they can be combined or reduced.
• Use Exponent Rules: Apply the laws of exponents to adjust the expression step by step.
• Double-Check Calculation: Always validate your simplification to prevent any arithmetic errors.

Becoming proficient in this process allows you to handle not just algebra but prepares you for more advanced topics.

Division of Powers

When dealing with exponents, dividing powers can be straightforward once you grasp the concept of removing repeated factors. As with our example, dividing powers involves the property that you subtract the exponents when the base is the same. That means from \(x^{14}/x^7\), we deducted 7 from 14, arriving at \(x^7\).

To ensure that this process is clear:

• Select Same Base: The division rule for exponents only works if the base remains consistent.
• Setup Clear Subtraction: Take the exponent from the numerator and subtract the exponent of the denominator \(x^{n-m}\).
• Simplify Further if Possible: If the result can be reduced or factored more, do so for a fully simplified form.

Understanding division of powers is critical, giving you the ease to answer problems involving complex expressions without getting lost in arithmetic.