Question

Write the expression in the form \(x^{n}\), assuming \(x \neq 0\) $$ \frac{x^{3} \cdot x^{4}}{x^{2}} $$

Short answer

Question: Simplify the expression \(\frac{x^3 \cdot x^4}{x^2}\) to the form \(x^n\). Answer: \(x^5\)

Step-by-step solution

Multiply the powers of x in the numerator

Remember that when multiplying powers of the same base, we add their exponents. So, in the numerator, we have: $$ x^3 \cdot x^4 = x^{3+4} $$

Simplify the expression in the numerator

Add the exponents in the previous step to get the simplified expression for the numerator: $$ x^{3+4} = x^7 $$ So the expression becomes: $$ \frac{x^7}{x^2} $$

Divide the powers of x in the fraction

Now, we need to divide the powers of x in the fraction. When dividing powers of the same base, we subtract their exponents. So, we have: $$ \frac{x^7}{x^2} = x^{7-2} $$

Simplify the expression

Subtract the exponents in the previous step to get the final expression: $$ x^{7-2} = x^5 $$ So the simplified expression in the form \(x^n\) is: $$ x^5 $$

Key concepts

Multiplying Exponents

Multiplying exponents is a common operation in algebra, especially when dealing with powers of the same base. When you encounter a scenario where you need to multiply exponents with the same base, the key rule is simple: add the exponents together. This is known as the Product of Powers Property.

For example, if you have two terms, like \( x^3 \) and \( x^4 \), both raised to the base \( x \), the result of multiplying these together is \( x^{3+4} = x^7 \). Here, the base remains the same, and the exponents are added.

This rule makes calculations with exponents more straightforward and helps simplify expressions efficiently. Always remember:

• Multiplying powers with the same base \( a^m \cdot a^n = a^{m+n} \)
• The base \( a \) must be the same for this rule to apply

Understanding this concept ensures that expressions are simplified correctly and systematically.

Dividing Exponents

Dividing exponents is another essential skill in algebra, especially when simplifying expressions. When dividing terms with the same base raised to different powers, the foundational rule to remember is to subtract the exponents. This principle is called the Quotient of Powers Property.

For instance, if you have an expression like \( \frac{x^7}{x^2} \), where both the numerator and the denominator have the same base \( x \), you subtract the exponents: \( x^{7-2} = x^5 \). This operation efficiently consolidates the expression into a single term with a new exponent.

Key points to keep in mind when dividing exponents:

• Division of powers with the same base \( a^m / a^n = a^{m-n} \)
• The subtraction occurs only for the exponents, not the base

Mastering these rules helps ensure accurate expressions in their simplest forms.

Simplifying Expressions

Simplifying expressions involves combining operations such as multiplying and dividing exponents to write an expression in its simplest form. It's about reducing an expression to an easy-to-understand format without altering its value.

Take the original expression \( \frac{x^{3} \cdot x^{4}}{x^{2}} \). First, multiply the exponents: \( x^3 \cdot x^4 = x^7 \). Then simplify the division: \( \frac{x^7}{x^2} = x^5 \). By applying the rules for multiplying and dividing exponents, the expression is reduced to \( x^5 \), a simplified form.

When simplifying expressions:

• Always follow the order of operations: first manage any multiplications/divisions, then any additions/subtractions of exponents
• Ensure that like terms, particularly with the same base, are simplified before moving on

Simplifying expressions not only makes them easier to work with but also prepares the groundwork for tackling more complex algebraic problems.