Problem 62 - Algebra 1: Concepts and Skills

Question

Simplify the expression. Use only positive exponents. $$x^{7} \cdot \frac{1}{x^{4}}$$

Step-by-Step Solution

Answer

The simplified expression with only positive exponents is \(x^{3}\).

Step 1: Identify the base and the exponents

We see that the base is 'x' in both parts of the expression. The exponents are 7 in \(x^{7}\) and 4 in \( \frac{1}{x^{4}}\).

Step 2: Apply the rule of exponents

According to the rule of exponents for division, when we divide like bases, we subtract the exponents. The expression then simplifies to \( x^{7 - 4} \).

Step 3: Simplify the expression

Now subtract the exponents to get the final answer. So, the simplified expression is \(x^{3}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables and constants, using operations like addition, subtraction, multiplication, and division. For instance, in the exercise "\(x^{7} \cdot \frac{1}{x^{4}}\)," we see an algebraic expression where 'x' is the variable, and it is used with exponents. This expression involves the multiplication of two components, which requires understanding of basic algebraic manipulation techniques.
When simplifying algebraic expressions, follow these steps:

• Identify the parts of the expression (variables, constants, operations).
• Combine like terms, which are terms with the same variable raised to the same power.
• Use rules of arithmetic and algebra to simplify.

Algebraic expressions are foundational in math as they allow us to represent real-world situations mathematically. Grasping their basic operations is essential for more advanced topics like calculus and algebraic equations.

Rule of Exponents

The rule of exponents is a set of guidelines for simplifying expressions with powers. It helps manage operations involving multiplication or division of expressions with the same base. In the case of the original exercise, understanding this rule is vital.
Key rules include:

• When multiplying like bases, add their exponents: \(a^m \times a^n = a^{m+n}\).
• When dividing like bases, subtract the exponents: \(a^m \div a^n = a^{m-n}\).
• An exponent of zero for any nonzero base is one: \(a^0 = 1\).
• A negative exponent indicates a reciprocal: \(a^{-n} = \frac{1}{a^n}\).

Here, the application of dividing like bases with the rule \(x^m \div x^n = x^{m-n}\) simplifies the problem to \(x^{7-4}\), resulting in \(x^3\). Mastery of these rules makes handling complex algebraic expressions much easier.

Exponent Subtraction

Exponent subtraction is applied while simplifying expressions with division involving powers. It stems from the rule of exponents where dividing similar bases results in subtracting their exponents.
In the exercise, we have \(x^{7}\) divided by \(x^{4}\). According to exponent rules, we subtract the exponent in the denominator from the one in the numerator: \(7 - 4\). This results in the simplified expression \(x^{3}\).
To apply exponent subtraction effectively, consider:

• Ensure the bases are the same before subtracting the exponents.
• Subtract the exponent of the denominator from the numerator.
• Express the result with the same base and the new exponent.

Understanding exponent subtraction is crucial, as it simplifies expressions and helps in solving algebraic equations efficiently. It is an essential tool in algebra that makes dealing with powers much more manageable.