Problem 22 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. $$ \frac{x^{5}}{x^{2}}=x^{2} $$

Step-by-Step Solution

Answer

The completed statement is \(\frac{x^{5}}{x^{2}}=x^{3}\).

Step 1: Recognize the properties of exponents

First, you need to identify that \(\frac{x^{5}}{x^{2}}\) involves the quotient of powers property of exponents. This property states that for any non-zero number a and any integers m and n, \(\frac{a^{m}}{a^{n}} = a^{m - n}\).

Step 2: Apply the properties of exponents

Next, apply the quotient of powers property by subtracting the exponent of the denominator, 2, from the exponent of the numerator, 5. This simplifies the equation to \(x^{5 - 2}\).

Step 3: Compute and Simplify

Complete the subtraction which results to \(x^{3}\).

Key Concepts

Quotient of Powers

Understanding the quotient of powers is fundamental in simplifying expressions that contain exponents. When you divide powers with the same base, you can simplify the expression by subtracting the exponent in the denominator from the one in the numerator. For example, \(\frac{x^{a}}{x^{b}}\) becomes \(x^{a-b}\). This operation is based on the idea that dividing by a number is the same as multiplying by its inverse.
In practical terms, imagine having five apples (\(x^{5}\)) and you give two away (\(x^{2}\)). You're left with three apples (\(x^{3}\)), which aligns with the arithmetic of exponents: \(x^{5-2}=x^{3}\). This visualization helps in remembering how exponents work when they are being divided.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form, making them easier to understand and work with. In the realm of exponents, this means using exponent rules to combine like terms or simplify complex expressions.
For example, simplifying the quotient of powers \(\frac{x^{5}}{x^{2}}\) requires recognizing that the x's are like terms and thus, can be simplified by subtracting the exponents. The result, \(x^{3}\), is the expression in its simplest form. Keeping expressions as simplified as possible is key when solving equations or working with algebraic operations, as it makes the subsequent steps more straightforward.

Exponent Rules

Exponent rules are shortcuts that help you work with powers efficiently. They are based on patterns observed when numbers are multiplied by themselves. The most common rules include the product of powers, power of a power, and quotient of powers, which we've just explored.

• The Product of Powers rule states that when you multiply powers with the same base, you add the exponents: \(x^{a} \cdot x^{b} = x^{a+b}\).
• The Power of a Power rule says that when raising a power to another power, you multiply the exponents: \( (x^{a})^{b} = x^{a\cdot b}\).
• The Quotient of Powers rule, which you've already seen, involves subtracting exponents when dividing powers with the same base: \(\frac{x^{a}}{x^{b}} = x^{a-b}\).

Remember, these rules apply to all numbers with exponents, not just variables. They are tools you can use to simplify and manipulate expressions in algebra.

Algebraic Operations

Algebraic operations encompass addition, subtraction, multiplication, and division, but in algebra, these are performed on variables as well as numbers. An essential part of algebraic operations is dealing with exponents through the exponent rules.
When you perform algebraic operations on expressions with exponents, you often need to simplify the expressions first. This is necessary to reduce the complexity and make the equation solvable. For instance, the example \(\frac{x^{5}}{x^{2}}\) initially looks like a division problem. Still, after using quotient of powers, the expression is simplified to multiplication of \(x\) by itself three times (\[x^3\]).
Remembering the order of operations—Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), often abbreviated as PEMDAS—is incredibly important in algebra to perform the steps correctly.