Solved Exercises on Dividing Powers with the Same Base in Grade 7

Master dividing powers with the same base: base, exponent, quotient rule, zero exponent, negative exponent through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Quotient Rule
Exercise 1
Calculate: \( \frac{4^7}{4^3} \)
Definition:

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

Division method:
  1. Identify the same base
  2. Subtract the exponent of the denominator from the exponent of the numerator
  3. Keep the same base
Original Expression
\(\frac{4^7}{4^3}\)
Same Base
\(4\)
Subtract Exponents
\(7 - 3 = 4\)
Result
\(4^4\)
Step 1: Identify the same base

Both numerator and denominator have base 4

Step 2: Apply the quotient rule

\(\frac{a^m}{a^n} = a^{m-n}\)

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

Step 3: Subtract the exponents

\(7 - 3 = 4\)

Step 4: Write the result

\(4^4\)

Step 5: Calculate if needed

\(4^4 = 256\)

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

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

Applied rules:

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

• Same base: Only applies when bases are identical

• Exponent subtraction: Subtract denominator exponent from numerator exponent

2 Negative Exponent Result
Exercise 2
Calculate: \( \frac{5^2}{5^6} \)
Definition:

Negative exponent: When numerator exponent is smaller, result is a negative exponent: \(\frac{a^m}{a^n} = a^{m-n}\) where \(m < n\)

Original Expression
\(\frac{5^2}{5^6}\)
Same Base
\(5\)
Subtract Exponents
\(2 - 6 = -4\)
Result
\(5^{-4}\)
Convert (Optional)
\(\frac{1}{5^4}\)
Step 1: Identify the same base

Both have base 5: numerator has \(5^2\), denominator has \(5^6\)

Step 2: Apply the quotient rule

\(\frac{a^m}{a^n} = a^{m-n}\)

\(\frac{5^2}{5^6} = 5^{2-6}\)

Step 3: Subtract the exponents

\(2 - 6 = -4\)

Step 4: Write the result

\(5^{-4}\)

Step 5: Convert to positive exponent (optional)

\(5^{-4} = \frac{1}{5^4} = \frac{1}{625}\)

\( \frac{5^2}{5^6} = 5^{-4} = \frac{1}{625} \)
Final answer:

\( \frac{5^2}{5^6} = 5^{-4} \)

Applied rules:

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

• Negative exponent: When \(m < n\), result is negative exponent

• Reciprocal form: \(a^{-n} = \frac{1}{a^n}\)

3 Zero Exponent
Exercise 3
Calculate: \( \frac{7^5}{7^5} \)
Definition:

Zero exponent rule: Any non-zero number to the power of 0 equals 1: \(a^0 = 1\) where \(a \neq 0\)

Original Expression
\(\frac{7^5}{7^5}\)
Same Base
\(7\)
Subtract Exponents
\(5 - 5 = 0\)
Result
\(7^0\)
Apply Zero Rule
\(1\)
Step 1: Identify the same base

Both numerator and denominator have base 7

Step 2: Apply the quotient rule

\(\frac{a^m}{a^n} = a^{m-n}\)

\(\frac{7^5}{7^5} = 7^{5-5}\)

Step 3: Subtract the exponents

\(5 - 5 = 0\)

Step 4: Write the result

\(7^0\)

Step 5: Apply the zero exponent rule

For any non-zero number \(a\), \(a^0 = 1\)

Therefore, \(7^0 = 1\)

\( \frac{7^5}{7^5} = 7^0 = 1 \)
Final answer:

\( \frac{7^5}{7^5} = 1 \)

Applied rules:

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

• Zero exponent: \(a^0 = 1\) (where \(a \neq 0\))

• Identity: Any non-zero number divided by itself equals 1

Rules and methods, laws,...
\( \frac{a^m}{a^n} = a^{m-n} \)
Quotient Rule for Exponents
Basic Quotient
\( \frac{a^m}{a^n} = a^{m-n} \)
Subtract exponents, keep base
Negative Result
\( \frac{a^m}{a^n} = a^{m-n} \) (if \(m < n\))
Results in negative exponent
Zero Exponent
\( \frac{a^n}{a^n} = a^0 = 1 \)
Any non-zero number to power 0 equals 1
Key definitions:

Power: An expression of the form \(a^n\) where \(a\) is the base and \(n\) is the exponent

Base: The number that is being multiplied repeatedly

Exponent: The number of times the base is multiplied by itself

Quotient rule: When dividing powers with the same base, subtract the exponents

Like bases: Powers that have identical base numbers

Exponent subtraction: The operation performed on exponents when dividing

Zero exponent: Any non-zero number raised to the power of 0 equals 1

Power division methods:
  1. Identify same bases: Ensure all terms have identical bases
  2. Apply quotient rule: Subtract denominator exponent from numerator exponent
  3. Handle special cases: Zero or negative results
  4. Simplify: Calculate if needed or leave in exponential form
Tip 1: Only apply this rule when bases are exactly the same
Tip 2: Subtract the denominator exponent from the numerator exponent
Tip 3: When numerator exponent is smaller, result is negative exponent
Tip 4: Remember: \(a^0 = 1\) for any \(a \neq 0\)
Common errors: Applying rule to different bases, subtracting in wrong order, forgetting sign rules, misapplying zero exponent rule.
Exam preparation: Practice all scenarios, memorize the quotient rule, watch for different bases.
Formulas to know by heart:

• Basic quotient rule: \(\frac{a^m}{a^n} = a^{m-n}\)

• Zero exponent: \(a^0 = 1\) (where \(a \neq 0\))

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

• Identity: \(\frac{a}{a} = 1\) (where \(a \neq 0\))

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

Solution: Exercises 4 to 5
4 Fractional Base
Exercise 4
Calculate: \( \frac{\left(\frac{1}{3}\right)^4}{\left(\frac{1}{3}\right)^2} \)
Definition:

Fractional base: The quotient rule applies to fractional bases too: \(\frac{\left(\frac{a}{b}\right)^m}{\left(\frac{a}{b}\right)^n} = \left(\frac{a}{b}\right)^{m-n}\)

Original Expression
\(\frac{\left(\frac{1}{3}\right)^4}{\left(\frac{1}{3}\right)^2}\)
Same Base
\(\frac{1}{3}\)
Subtract Exponents
\(4 - 2 = 2\)
Result
\(\left(\frac{1}{3}\right)^2\)
Simplify
\(\frac{1}{9}\)
Step 1: Identify the same base

Both numerator and denominator have the same fractional base: \(\frac{1}{3}\)

Step 2: Apply the quotient rule

\(\frac{\left(\frac{1}{3}\right)^4}{\left(\frac{1}{3}\right)^2} = \left(\frac{1}{3}\right)^{4-2}\)

Step 3: Subtract the exponents

\(4 - 2 = 2\)

Step 4: Write the result

\(\left(\frac{1}{3}\right)^2\)

Step 5: Calculate the power

\(\left(\frac{1}{3}\right)^2 = \frac{1^2}{3^2} = \frac{1}{9}\)

\( \frac{\left(\frac{1}{3}\right)^4}{\left(\frac{1}{3}\right)^2} = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \)
Final answer:

\( \frac{\left(\frac{1}{3}\right)^4}{\left(\frac{1}{3}\right)^2} = \frac{1}{9} \)

Applied rules:

• Quotient rule: Applies to fractional bases too

• Same base: Both terms have base \(\frac{1}{3}\)

• Exponent subtraction: Subtract exponents when dividing

5 Variable Base
Exercise 5
Calculate: \( \frac{x^8}{x^3} \) where \(x \neq 0\)
Definition:

Variable base: The quotient rule works with variables too: \(\frac{x^m}{x^n} = x^{m-n}\)

Original Expression
\(\frac{x^8}{x^3}\)
Same Base
\(x\)
Subtract Exponents
\(8 - 3 = 5\)
Result
\(x^5\)
Step 1: Identify the same base

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

Step 2: Apply the quotient rule

\(\frac{x^m}{x^n} = x^{m-n}\)

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

Step 3: Subtract the exponents

\(8 - 3 = 5\)

Step 4: Write the result

\(x^5\)

Step 5: Note restrictions

Condition: \(x \neq 0\) (to avoid division by zero)

\( \frac{x^8}{x^3} = x^5 \)
Final answer:

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

Applied rules:

• Quotient rule: Works with variable bases

• Variable exponents: Subtract exponents algebraically

• General form: \(\frac{x^m}{x^n} = x^{m-n}\)

Key Concepts: Laws, Methods, Rules, Definitions
\( \frac{a^m}{a^n} = a^{m-n} \)
Fundamental Quotient Rule
Key definitions:

Power: An expression of the form \(a^n\) where \(a\) is the base and \(n\) is the exponent

Base: The number that is being multiplied repeatedly

Exponent: The number of times the base is multiplied by itself

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

Like bases: Powers that have identical base numbers

Exponent subtraction: The operation performed on exponents when dividing

Algebraic expression: Using variables as bases in power expressions

Complete division methodology:
  1. Identify like bases: Check that all terms have identical bases
  2. Verify conditions: Ensure bases are exactly the same
  3. Apply quotient rule: Subtract denominator exponent from numerator exponent
  4. Keep base: The result uses the same base as the original terms
  5. Simplify: Calculate if needed or leave in exponential form
  6. Check restrictions: Note any domain restrictions
  7. Verify: Confirm by expanding if necessary
Tip 1: The rule only works when bases are exactly the same
Tip 2: Subtract exponents even when they're negative
Tip 3: This rule applies to numbers, variables, and fractions
Tip 4: Remember: different bases cannot use this rule
Common errors: Applying to different bases, subtracting in wrong order, forgetting to check for zero base.
Exam preparation: Practice with various base types, memorize the quotient rule, watch for disguised same bases.
Formulas to know by heart:

• Basic quotient rule: \(\frac{a^m}{a^n} = a^{m-n}\)

• Zero exponent: \(a^0 = 1\) (where \(a \neq 0\))

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

• Fractional base: \(\frac{\left(\frac{a}{b}\right)^m}{\left(\frac{a}{b}\right)^n} = \left(\frac{a}{b}\right)^{m-n}\)

• Variable base: \(\frac{x^m}{x^n} = x^{m-n}\)

• Identity: \(\frac{a}{a} = 1\) (where \(a \neq 0\))

Exercise with Visualization: Power Division Patterns
Exercise 6: Division of Powers with Same Base
Observe these division patterns:
\( \frac{2^5}{2^2} = 2^3 \)
\( \frac{3^4}{3^1} = 3^3 \)
\( \frac{5^6}{5^4} = 5^2 \)
\( \frac{4^3}{4^3} = 4^0 = 1 \)

Analysis: The chart shows how exponents subtract when dividing powers with the same base.

  • \( \frac{2^5}{2^2} = 2^{5-2} = 2^3 \)
  • \( \frac{3^4}{3^1} = 3^{4-1} = 3^3 \)
  • \( \frac{5^6}{5^4} = 5^{6-4} = 5^2 \)
  • \( \frac{4^3}{4^3} = 4^{3-3} = 4^0 = 1 \)

Questions & Answers

Question: Why do we subtract the exponents when dividing powers with the same base? It seems backwards!

Answer: This makes perfect sense when you think about what division means!

Let's look at \(\frac{2^5}{2^3}\):

  • \(2^5 = 2 \times 2 \times 2 \times 2 \times 2\) (five 2's)
  • \(2^3 = 2 \times 2 \times 2\) (three 2's)
  • So \(\frac{2^5}{2^3} = \frac{2 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2}\)
  • We can cancel out three 2's from top and bottom
  • This leaves us with two 2's: \(2 \times 2 = 2^2\)

Since we started with 5 twos and cancelled 3 of them, we end up with \(5 - 3 = 2\) twos remaining.

That's why the rule is: subtract the exponents when dividing like bases!

Question: What happens if the bases are different? Can I still subtract the exponents?

Answer: No! The quotient rule \(\frac{a^m}{a^n} = a^{m-n}\) only works when the bases are identical.

Examples:

  • \(\frac{2^5}{2^3} = 2^{5-3} = 2^2\) ✓ (same base)
  • \(\frac{2^5}{3^3} \neq \left(\frac{2}{3}\right)^{5-3}\) ✗ (different bases)
  • \(\frac{2^5}{3^3} = \frac{32}{27}\) (calculate separately)

When bases are different, you must evaluate each power separately and then divide the results.

The key is: Same base → subtract exponents. Different bases → calculate separately.

Question: How do I know if my division of powers is correct?

Answer: Here are several ways to verify your power division:

  1. Check same base: Ensure all terms have identical bases
  2. Verify exponent subtraction: Make sure you subtracted correctly
  3. Expand to check: For small exponents, expand and divide manually
  4. Reverse operation: Multiply your answer by the denominator to get the numerator
  5. Estimation: Check if your answer is reasonable

Example verification: For \(\frac{3^4}{3^2} = 3^2\)

  • Same base check: Both have base 3 ✓
  • Exponent subtraction: \(4 - 2 = 2\) ✓
  • Manual check: \(\frac{81}{9} = 9\) and \(3^2 = 9\) ✓
  • Reverse: \(3^2 \times 3^2 = 3^4\) ✓

Always double-check that you're applying the rule only to terms with identical bases!