Problem 81 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ \frac{x^{n+2} x^{3}}{x^{4} x^{n}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$ \frac{x^{n+2}x^3}{x^4x^n} $$ assuming x is nonzero and exponents are whole numbers. Answer: x

Step 1: Apply the product rule in the numerator

Since the bases (x) are the same, we can add the exponents in the numerator to simplify: $$ x^{n+2}x^3 = x^{(n+2) + 3} = x^{n+5} $$

Step 2: Apply the product rule in the denominator

Similarly, we can add the exponents in the denominator to simplify: $$ x^4x^n = x^{4+n} $$

Step 3: Apply the quotient rule

Now that we have simplified expressions in the numerator and denominator, we can use the quotient rule by subtracting the exponent in the denominator from the exponent in the numerator: $$ \frac{x^{n+5}}{x^{4+n}} = x^{(n+5) - (4+n)} = x^{5 - 4} = x^{1} $$

Step 4: Final answer

The simplified expression is: $$ x^1 $$ or simply $$ x $$

Key Concepts

Product Rule of Exponents

The product rule of exponents is extremely helpful when you are multiplying expressions with the same base. The rule says that when you multiply two powers with the same base, you simply add the exponents together. This happens because you essentially have repeated multiplication, so you count up all the exponents together.
For example, if you multiply \( x^a \times x^b \), you get \( x^{a+b} \).

Applying this rule helps to simplify expressions quickly and easily. It's a basic yet powerful tool that develops your algebra skills.

• Always make sure that the bases are identical before using this rule.
• Only the exponents are affected; the base stays the same during the process.

When working with expressions like \( x^{n+2} \times x^3 \), using the product rule, you add the exponents: \( (n+2) + 3 = n+5 \). This results in \( x^{n+5} \). Knowing this rule makes handling multiple expressions a breeze!

Quotient Rule of Exponents

When dealing with division of expressions with the same base, the quotient rule of exponents becomes your go-to tool. This rule states that when you divide two powers with the same base, you subtract the exponent in the denominator from the exponent in the numerator.
Imagine simplifying \( \frac{x^c}{x^d} \); using the quotient rule, this turns into \( x^{c-d} \). This rule simplifies expressions by reducing the number of similar base powers.

The simplified process makes your algebraic expressions cleaner and easier to manage, just as in our main example:

• Verify that the bases match. If not, you can't apply this rule.
• Subtract exponents always—from the numerator down to the denominator, not the other way around.

Applying this to \( \frac{x^{n+5}}{x^{4+n}} \), you subtract \( (4+n) \) from \( (n+5) \) resulting in \( x^{5-4} = x^{1} \). This illustrates how subtraction of exponents is used in the quotient rule.

Algebraic Simplification

Algebraic simplification is the process of making an expression easier to work with or understand without changing its value. It involves applying various mathematical rules and techniques to reduce complexity.
In the context of exponents, simplification often means using the product and quotient rules to condense terms into the smallest possible expression.

This process allows students to handle complex algebraic expressions confidently and is crucial as a foundation for solving more advanced mathematical problems.

• Focus on combining like terms, which are terms with identical bases.
• Apply exponent rules consistently so that each step leads to a greater reduction in complexity.

Take our original problem, \( \frac{x^{n+2} x^3}{x^4 x^n} \). By applying first the product rule and then the quotient rule, we've simplified a seemingly complex expression down to just \( x \). Algebraic simplification is a key skill that takes practice but becomes intuitive with regular attention.