Problem 36 - Algebra and Trigonometry

Question

\(29-46\) Simplify each expression. $$ \frac{x^{6}}{x^{10}} $$

Step-by-Step Solution

Answer

\( \frac{1}{x^4} \)

Step 1: Understand the Expression

We want to simplify the expression \( \frac{x^6}{x^{10}} \). This is a division of exponents with the same base \(x\).

Step 2: Apply the Quotient Rule for Exponents

According to the quotient rule of exponents, when we divide two expressions with the same base, we subtract the exponents: \( \frac{x^a}{x^b} = x^{a-b} \).

Step 3: Subtract the Exponents

Apply the quotient rule to our expression: \( \frac{x^6}{x^{10}} = x^{6-10} \).

Step 4: Simplify the Exponent

Subtract 10 from 6 to get \( -4 \). So, \( x^{6-10} = x^{-4} \).

Step 5: Express as a Positive Exponent

Remember that \( x^{-n} = \frac{1}{x^n} \). Therefore, \( x^{-4} = \frac{1}{x^4} \).

Key Concepts

Quotient Rule for Exponents

The quotient rule for exponents is a fundamental rule that helps simplify expressions involving division. When you have the same base raised to different powers and you divide them, you can apply the quotient rule:

• For any expression of the form \(\frac{x^a}{x^b}\), where \(x\) is the common base, the quotient rule states you simply subtract the exponent of the denominator from the exponent of the numerator.
• The rule simplifies the division of such exponential expressions by focusing on the exponents alone, not the base.

Let’s see an example: to simplify \(\frac{x^6}{x^{10}}\), subtract the exponent 10 from 6, giving you \(x^{6-10} = x^{-4}\). This operation is convenient in algebra because it allows you to manage large expressions efficiently. If the base was different, you could not apply this rule. Always make sure the bases are the same before applying the quotient rule.

Negative Exponents

Negative exponents might seem tricky at first, but they just mean you take the reciprocal of the base raised to the corresponding positive exponent. Imagine you have an expression like \(x^{-4}\). What this tells you is to invert the base:

• \(x^{-n} = \frac{1}{x^n}\) for any base \(x\) and exponent \(n\).
• This is a quick way to convert negative exponents into positive ones, a necessary step when simplifying expressions.

In our example, where we ended up with \(x^{-4}\), this transforms into \(\frac{1}{x^4}\). This transformation is useful because it helps you express the result in a more standard form, avoiding negative exponents in your final answer.

Exponent Rules

Exponent rules are key to navigating expressions involving powers. They provide shortcuts that simplify complex algebraic expressions.Some of the primary rules include:

• Product Rule: When multiplying powers with the same base, add the exponents: \(x^a \cdot x^b = x^{a+b}\).
• Power Rule: Raising a power to another power? Multiply the exponents: \((x^a)^b = x^{a\cdot b}\).
• Quotient Rule: As discussed before, divide powers by subtracting exponents: \(\frac{x^a}{x^b} = x^{a-b}\).
• Zero Exponent Rule: Any non-zero base raised to the power of zero is 1: \(x^0 = 1\).

Mastering these rules is crucial as they often combine in complex problems.Understanding how these rules interrelate can help you simplify and solve equations more efficiently.

Simplification Steps

Simplifying expressions with exponents involves a set method, making it easy to follow.Here's a step-by-step approach:

• Identify Common Base: Check if the expression's terms share the same base.
• Apply Simplification Rules: Choose the appropriate exponent rule based on operation: multiplication, division, or power of a power.
• Simplify the Exponents: For division like \(\frac{x^6}{x^{10}}\), subtract exponents to find the new power, finally resulting in \(x^{-4}\).
• Convert Negative Exponents (if any): Use the negative exponent rule to rewrite in positive form, giving \(\frac{1}{x^4}\).
• Ensure Expression is Fully Simplified: Check that all exponents are positive and reduced.

Following these steps ensures thorough and correct simplification of expressions including exponents, eliminating common errors and confusion in solving such problems.