Question

Write the expression in the form \(x^{n}\), assuming \(x \neq 0\) $$ \frac{x^{7}}{x^{4}} \cdot \frac{x^{5}}{x} $$

Short answer

Question: Simplify the following expression and write it in the form \(x^n\): \(\frac{x^7}{x^4} \cdot \frac{x^5}{x^1}\) Solution: The simplified expression is \(x^7\).

Step-by-step solution

Simplify the quotients using exponent division rule

According to the exponent division rule: \(a^{m} / a^{n} = a^{(m-n)}\). Apply this rule to both quotients in the expression: $$ \frac{x^7}{x^4} = x^{(7-4)} = x^3 \\ \frac{x^5}{x^1} = x^{(5-1)} = x^4 $$ Now, our expression becomes: $$ x^3 \cdot x^4 $$ #Step 2: Apply the property of exponent for products#

Simplify the product using exponent multiplication rule

According to the exponent multiplication rule, when multiplying two expressions with the same base, the exponents are added: \(a^{m} \cdot a^{n} = a^{(m+n)}\). Apply this rule to the product in our simplified expression: $$ x^3 \cdot x^4 = x^{(3+4)} = x^7 $$ #Solution# The simplified expression in the form \(x^n\) is: $$ x^7 $$

Key concepts

Exponent Division Rule

When dealing with expressions that involve dividing similar bases, the exponent division rule simplifies the task immensely. According to this rule, when you divide two numbers or variables that have the same base, you subtract the exponent of the denominator from the exponent of the numerator. This can be expressed mathematically as \(a^{m} / a^{n} = a^{(m-n)}\).

Let's consider an example with the expression \(\frac{x^7}{x^4}\). Here, both the numerator and the denominator have the base 'x'. Using the exponent division rule, you would subtract the exponent 4 in the denominator from the exponent 7 in the numerator, resulting in \(x^{(7-4)} = x^3\).

The same principle applies to \(\frac{x^5}{x^1}\), where \(1\) is often implied but not written. Subtracting \(1\) from \(5\) gives \(x^{(5-1)} = x^4\). By understanding and applying this rule, expressions that could initially seem complex are made much simpler.

Exponent Multiplication Rule

Multiplying expressions with the same base requires using the exponent multiplication rule, which states that you add the exponents together. This can be formulated as \(a^{m} \cdot a^{n} = a^{(m+n)}\). It simplifies calculations especially when variables with exponents need to be multiplied together.

Imagine you have the expression \(x^3 \cdot x^4\). Both terms share the base 'x'. According to the rule, you simply add the exponents, \(3\) and \(4\), to get \(x^{(3+4)} = x^7\).

By applying the exponent multiplication rule, you can merge powers efficiently without individually multiplying each term. It is crucial when simplifying large expressions, as it saves time and minimizes errors.

Simplifying Expressions

Simplifying expressions is a vital mathematical skill that involves reducing an expression to its simplest form. It often involves applying rules for exponents, such as division and multiplication, to achieve a more straightforward version.

In problems like \(\frac{x^{7}}{x^{4}} \cdot \frac{x^{5}}{x}\), the goal is to simplify everything down to one expression in the form \(x^n\). First, use the exponent division rule to handle the fractions, turning them into \(x^3\) and \(x^4\). Then, apply the exponent multiplication rule to find the product, leading to the final form \(x^7\).

Ensuring that expressions are simplified helps in many areas of math, such as solving equations and evaluating outcomes. It makes understanding the relationships within mathematical problems clearer and results easier to handle. Employing these exponent rules effectively reduces complexities and enhances problem-solving efficiency.