Solved Exercises on Dividing Powers with Same Base in Grade 8

Master dividing powers with same base: basic quotient rule, multiple terms, coefficients, complex expressions, and scientific notation through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Quotient Rule
Exercise 1
Simplify:
\(\frac{x^7}{x^3}\)
Definition:

Quotient Rule: When dividing powers with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\)

Quotient Rule Method:
  1. Verify that bases are identical
  2. Subtract the exponent in the denominator from the exponent in the numerator
  3. Keep the same base
Expression
\(\frac{x^7}{x^3}\)
Apply Quotient Rule
\(x^{7-3}\)
Subtract Exponents
\(x^4\)
Step 1: Verify same base

Both numerator and denominator have the same base: \(x\)

Step 2: Apply the quotient rule

\(\frac{x^7}{x^3} = x^{7-3}\)

Step 3: Calculate the difference of exponents

\(7 - 3 = 4\)

Step 4: Write the final answer

\(x^4\)

\(\frac{x^7}{x^3} = x^4\)
Final answer:

\(\frac{x^7}{x^3} = x^4\)

Applied rules:

• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)

• Same Base Requirement: Only applies when bases are identical

• Exponent Subtraction: Subtract denominator exponent from numerator exponent

• Base Preservation: Keep the same base

2 Denominator Exponent Larger
Exercise 2
Simplify:
\(\frac{x^2}{x^5}\)
Definition:

Quotient Rule with Negative Result: When the denominator exponent is larger, the result has a negative exponent

Expression
\(\frac{x^2}{x^5}\)
Apply Quotient Rule
\(x^{2-5}\)
Subtract Exponents
\(x^{-3}\)
Convert to Positive
\(\frac{1}{x^3}\)
Step 1: Verify same base

Both numerator and denominator have the same base: \(x\)

Step 2: Apply the quotient rule

\(\frac{x^2}{x^5} = x^{2-5}\)

Step 3: Calculate the difference of exponents

\(2 - 5 = -3\)

Step 4: Convert to positive exponent (optional)

\(x^{-3} = \frac{1}{x^3}\)

\(\frac{x^2}{x^5} = x^{-3} = \frac{1}{x^3}\)
Final answer:

\(\frac{x^2}{x^5} = x^{-3}\) or \(\frac{1}{x^3}\)

Applied rules:

• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)

• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)

• Exponent Subtraction: May result in negative exponents

3 With Coefficients
Exercise 3
Simplify:
\(\frac{12x^6}{4x^2}\)
Definition:

Quotients with Coefficients: Divide coefficients separately, then apply quotient rule to like bases

Expression
\(\frac{12x^6}{4x^2}\)
Separate Coefficients and Variables
\(\frac{12}{4} \cdot \frac{x^6}{x^2}\)
Divide Coefficients
\(3 \cdot \frac{x^6}{x^2}\)
Apply Quotient Rule
\(3x^4\)
Step 1: Separate coefficients and variables

\(\frac{12x^6}{4x^2} = \frac{12}{4} \cdot \frac{x^6}{x^2}\)

Step 2: Divide the coefficients

\(\frac{12}{4} = 3\)

Step 3: Apply the quotient rule to variables

\(\frac{x^6}{x^2} = x^{6-2} = x^4\)

Step 4: Combine the results

\(3 \cdot x^4 = 3x^4\)

\(\frac{12x^6}{4x^2} = 3x^4\)
Final answer:

\(\frac{12x^6}{4x^2} = 3x^4\)

Applied rules:

• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)

• Coefficient Division: Divide coefficients separately

• Like Bases Only: Apply quotient rule only to identical bases

Dividing Powers with Same Base Rules and Methods
\(\frac{a^m}{a^n} = a^{m-n}\)
Quotient Rule
Basic Quotient Rule
\(\frac{a^m}{a^n} = a^{m-n}\)
Divide powers with same base by subtracting exponents
Negative Result
\(\frac{a^m}{a^n} = a^{m-n}\) (if \(m < n\))
May result in negative exponents
With Coefficients
\(\frac{ka^m}{la^n} = \frac{k}{l}a^{m-n}\)
Divide coefficients separately
Negative Exponents
\(\frac{a^{-m}}{a^n} = a^{-m-n}\)
Rule applies with negative exponents
Zero Exponent
\(\frac{a^m}{a^m} = a^0 = 1\)
Equal exponents result in 1
Fractional Exponents
\(\frac{a^{m/n}}{a^{p/q}} = a^{m/n - p/q}\)
Rule extends to fractional exponents
Key Concepts: The quotient rule only applies when dividing powers with identical bases. The base remains unchanged while exponents are subtracted.
Important Note: If the denominator exponent is larger than the numerator exponent, the result will have a negative exponent.
Tip 1: Always check that bases are identical before applying the quotient rule.
Tip 2: When dividing terms with coefficients, handle coefficients and variables separately.
Tip 3: Remember: Quotient Rule = Subtract exponents (numerator minus denominator).
Solution: Exercises 4 to 5
4 Complex Expression with Same Base
Exercise 4
Simplify:
\(\frac{15x^8 \cdot 2x^3}{5x^4}\)
Definition:

Complex Quotients: Apply product rule in numerator first, then apply quotient rule

Original
\(\frac{15x^8 \cdot 2x^3}{5x^4}\)
Multiply Numerator
\(\frac{(15 \cdot 2)(x^8 \cdot x^3)}{5x^4}\)
Simplify Numerator
\(\frac{30x^{11}}{5x^4}\)
Separate Coefficients
\(\frac{30}{5} \cdot \frac{x^{11}}{x^4}\)
Final
\(6x^7\)
Step 1: Apply product rule in numerator

\(15x^8 \cdot 2x^3 = (15 \cdot 2)(x^8 \cdot x^3) = 30x^{8+3} = 30x^{11}\)

Step 2: Rewrite expression

\(\frac{30x^{11}}{5x^4}\)

Step 3: Separate coefficients and variables

\(\frac{30}{5} \cdot \frac{x^{11}}{x^4}\)

Step 4: Apply quotient rule

\(\frac{30}{5} = 6\) and \(\frac{x^{11}}{x^4} = x^{11-4} = x^7\)

Step 5: Combine results

\(6 \cdot x^7 = 6x^7\)

\(\frac{15x^8 \cdot 2x^3}{5x^4} = 6x^7\)
Final answer:

\(\frac{15x^8 \cdot 2x^3}{5x^4} = 6x^7\)

Applied rules:

• Product Rule: \(a^m \cdot a^n = a^{m+n}\)

• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)

• Coefficient Division: Divide coefficients separately

5 Scientific Notation with Same Base
Exercise 5
Divide and express in scientific notation:
\(\frac{8 \times 10^7}{2 \times 10^3}\)
Definition:

Scientific Notation: Numbers expressed as \(a \times 10^n\) where \(1 \leq a < 10\) and \(n\) is an integer

Original
\(\frac{8 \times 10^7}{2 \times 10^3}\)
Separate Coefficients and Powers
\(\frac{8}{2} \cdot \frac{10^7}{10^3}\)
Divide Coefficients
\(4 \cdot \frac{10^7}{10^3}\)
Apply Quotient Rule
\(4 \times 10^4\)
Step 1: Separate coefficients and powers of 10

\(\frac{8 \times 10^7}{2 \times 10^3} = \frac{8}{2} \cdot \frac{10^7}{10^3}\)

Step 2: Divide the coefficients

\(\frac{8}{2} = 4\)

Step 3: Apply quotient rule to powers of 10

\(\frac{10^7}{10^3} = 10^{7-3} = 10^4\)

Step 4: Combine results

\(4 \times 10^4\)

\(\frac{8 \times 10^7}{2 \times 10^3} = 4 \times 10^4\)
Final answer:

\(\frac{8 \times 10^7}{2 \times 10^3} = 4 \times 10^4\)

Applied rules:

• Separation Property: Separate coefficients and powers

• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)

• Scientific Notation: Express result properly

Complete Guide: Dividing Powers with Same Base, Rules, Methods, and Applications
\(\frac{a^m}{a^n} = a^{m-n}\)
Fundamental Rule
Key definitions:

Base: The number or variable being raised to a power

Exponent: The number indicating how many times the base is multiplied by itself

Quotient Rule: The rule for dividing powers with the same base

Coefficient: The numerical factor in front of a variable term

Complete methodology:
  1. Identify the structure: Confirm that both terms have the same base
  2. Separate coefficients: Handle numerical coefficients separately from variables
  3. Apply the quotient rule: Subtract the denominator exponent from the numerator exponent
  4. Perform arithmetic: Calculate differences of exponents and quotients of coefficients
  5. Simplify: Combine the results into a single expression
Tip 1: The quotient rule is \(\frac{a^m}{a^n} = a^{m-n}\) - subtract exponents, don't divide them!
Tip 2: When dividing terms, always handle coefficients and variables separately.
Tip 3: If the denominator exponent is larger, you'll get a negative exponent: \(\frac{a^m}{a^n} = a^{m-n}\) where \(m < n\) gives \(a^{negative}\).
Tip 4: Always verify that bases are identical before applying the quotient rule.
Common errors: Subtracting bases instead of exponents, applying the rule to different bases, forgetting to handle coefficients separately, confusing numerator and denominator exponents.
Exam preparation: Practice with various exponent scenarios, master coefficient handling, understand when the rule applies and when it doesn't.
Essential rules to memorize:

• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)

• With Coefficients: \(\frac{ka^m}{la^n} = \frac{k}{l}a^{m-n}\)

• Product Rule: \(a^m \cdot a^n = a^{m+n}\)

• Power Rule: \((a^m)^n = a^{mn}\)

• Zero Rule: \(a^0 = 1\) (when \(a \neq 0\))

• Negative Rule: \(a^{-n} = \frac{1}{a^n}\)

Exercise with Visualization: Dividing Powers Functions
Exercise 6: Dividing Powers Behavior
Consider the following functions:
\(f_1(x) = \frac{x^5}{x^2} = x^3\)
\(f_2(x) = \frac{x^7}{x^4} = x^3\)
\(f_3(x) = \frac{x^8}{x^5} = x^3\)

Analysis: The chart shows how dividing powers with same base results in polynomial functions with subtracted exponents.

  • \(f_1(x) = \frac{x^5}{x^2} = x^3\) (Quotient rule: subtract exponents)
  • \(f_2(x) = \frac{x^7}{x^4} = x^3\) (Different inputs, same result)
  • \(f_3(x) = \frac{x^8}{x^5} = x^3\) (Consistent pattern)

Questions & Answers

Question: Why do we subtract the exponents when dividing powers with the same base? Why don't we divide them?

Answer: Let's look at what division of powers actually means:

\(\frac{x^5}{x^2}\) means:

\(\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x} = x \cdot x \cdot x = x^3\)

We're counting how many times the base appears in the numerator compared to the denominator. The numerator has 5 x's and the denominator has 2 x's, so we're left with \(5 - 2 = 3\) x's in the numerator.

The quotient rule represents cancellation of common factors, which subtracts from the total count of how many times the base appears.

Dividing the exponents would be incorrect because it doesn't represent the actual division of the terms.

Question: What happens if the denominator exponent is larger than the numerator exponent? Like in \(\frac{x^2}{x^5}\)?

Answer: When the denominator exponent is larger, you get a negative exponent:

For \(\frac{x^2}{x^5}\):

  1. Apply the quotient rule: \(\frac{x^2}{x^5} = x^{2-5}\)
  2. Calculate the difference: \(2 - 5 = -3\)
  3. Result: \(x^{-3}\)
  4. Convert to positive exponent if needed: \(x^{-3} = \frac{1}{x^3}\)

This makes sense because if you think about it:

\(\frac{x^2}{x^5} = \frac{x \cdot x}{x \cdot x \cdot x \cdot x \cdot x} = \frac{1}{x \cdot x \cdot x} = \frac{1}{x^3}\)

So the quotient rule correctly gives us \(x^{-3} = \frac{1}{x^3}\).

Both forms are mathematically equivalent!

Question: How does dividing powers with same base relate to scientific notation? Why is this important?

Answer: Dividing powers with same base is fundamental to scientific notation calculations:

Examples in science:

  • Density calculations: \(\frac{6 \times 10^3}{2 \times 10^1} = 3 \times 10^2\)
  • Speed calculations: \(\frac{\text{distance}}{\text{time}}\) often involves powers of 10
  • Chemistry concentration: \(\frac{\text{moles}}{\text{volume}}\) calculations

For \(\frac{9 \times 10^6}{3 \times 10^2}\):

  1. Separate coefficients and powers: \(\frac{9}{3} \cdot \frac{10^6}{10^2}\)
  2. Divide coefficients: \(\frac{9}{3} = 3\)
  3. Apply quotient rule to powers: \(\frac{10^6}{10^2} = 10^{6-2} = 10^4\)
  4. Result: \(3 \times 10^4\)

This is crucial for scientific calculations because it allows us to handle very large or very small numbers efficiently while maintaining precision.

Understanding this rule helps scientists and engineers work with measurements across different scales!

Question: What happens if I have negative exponents in division? Like \(\frac{x^{-3}}{x^5}\)?

Answer: The quotient rule works the same way with negative exponents:

For \(\frac{x^{-3}}{x^5}\):

  1. Apply the quotient rule: \(\frac{x^{-3}}{x^5} = x^{(-3)-5}\)
  2. Subtract the exponents: \((-3) - 5 = -8\)
  3. Result: \(x^{-8}\)
  4. Convert to positive exponent if needed: \(x^{-8} = \frac{1}{x^8}\)

Key points:

  • The quotient rule \(\frac{a^m}{a^n} = a^{m-n}\) applies to negative exponents
  • Follow the rules for subtracting signed numbers
  • Positive - Negative = Add the absolute values
  • Negative - Positive = More negative

So \(\frac{x^{-3}}{x^5} = x^{-8}\) while \(\frac{x^3}{x^{-5}} = x^{3-(-5)} = x^8\)

Question: How do I know when to use the quotient rule versus other exponent rules? Sometimes I get confused about which rule to apply.

Answer: Here's how to identify which rule to use:

Look for these patterns:

  • Quotient Rule: \(\frac{a^m}{a^n}\) - Same base divided
  • Product Rule: \(a^m \cdot a^n\) - Same base multiplied together
  • Power Rule: \((a^m)^n\) - A power raised to another power
  • Power of Product: \((ab)^n\) - Multiple factors raised to a power

For complex expressions, identify the operation first:

  1. If you see division with same base: Use quotient rule (subtract exponents)
  2. If you see multiplication with same base: Use product rule (add exponents)
  3. If you see a power raised to another power: Use power rule (multiply exponents)

Example: \(\frac{x^5 \cdot x^2}{x^3}\)

  • First: Apply product rule to numerator: \(x^5 \cdot x^2 = x^7\)
  • Second: Apply quotient rule: \(\frac{x^7}{x^3} = x^{7-3} = x^4\)

Always identify the operation and base relationship first, then apply the appropriate rule!