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

Master multiplying powers with the same base: base, exponent, product rule through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Product Rule
Exercise 1
Calculate: \( 2^3 \times 2^4 \)
Definition:

Product rule: When multiplying powers with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\)

Multiplication method:
  1. Identify the same base
  2. Add the exponents
  3. Keep the same base
Original Expression
\(2^3 \times 2^4\)
Same Base
\(2\)
Add Exponents
\(3 + 4 = 7\)
Result
\(2^7\)
Step 1: Identify the same base

Both terms have base 2: \(2^3\) and \(2^4\)

Step 2: Apply the product rule

\(a^m \times a^n = a^{m+n}\)

\(2^3 \times 2^4 = 2^{3+4}\)

Step 3: Add the exponents

\(3 + 4 = 7\)

Step 4: Write the result

\(2^7\)

\( 2^3 \times 2^4 = 2^7 \)
Final answer:

\( 2^3 \times 2^4 = 2^7 \)

Applied rules:

• Product rule: \(a^m \times a^n = a^{m+n}\)

• Same base: Only applies when bases are identical

• Exponent addition: Add the exponents, keep the base

2 Negative Exponent
Exercise 2
Calculate: \( 5^2 \times 5^{-3} \)
Definition:

Negative exponent: \(a^{-n} = \frac{1}{a^n}\) and the product rule still applies

Original Expression
\(5^2 \times 5^{-3}\)
Same Base
\(5\)
Add Exponents
\(2 + (-3) = -1\)
Result
\(5^{-1}\)
Step 1: Identify the same base

Both terms have base 5: \(5^2\) and \(5^{-3}\)

Step 2: Apply the product rule

\(a^m \times a^n = a^{m+n}\)

\(5^2 \times 5^{-3} = 5^{2+(-3)}\)

Step 3: Add the exponents

\(2 + (-3) = 2 - 3 = -1\)

Step 4: Write the result

\(5^{-1}\)

Step 5: Simplify if needed

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

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

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

Applied rules:

• Product rule: \(a^m \times a^n = a^{m+n}\) (works with negative exponents)

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

• Integer addition: \(2 + (-3) = -1\)

3 Multiple Terms
Exercise 3
Calculate: \( 3^2 \times 3^5 \times 3^1 \)
Definition:

Extended product rule: For multiple terms with same base, add all exponents: \(a^m \times a^n \times a^p = a^{m+n+p}\)

Original Expression
\(3^2 \times 3^5 \times 3^1\)
Same Base
\(3\)
Add All Exponents
\(2 + 5 + 1 = 8\)
Result
\(3^8\)
Step 1: Identify the same base

All terms have base 3: \(3^2\), \(3^5\), and \(3^1\)

Step 2: Apply the extended product rule

\(a^m \times a^n \times a^p = a^{m+n+p}\)

\(3^2 \times 3^5 \times 3^1 = 3^{2+5+1}\)

Step 3: Add all exponents

\(2 + 5 + 1 = 8\)

Step 4: Write the result

\(3^8\)

Step 5: Calculate if needed

\(3^8 = 6561\)

\( 3^2 \times 3^5 \times 3^1 = 3^8 = 6561 \)
Final answer:

\( 3^2 \times 3^5 \times 3^1 = 3^8 \)

Applied rules:

• Extended product rule: \(a^m \times a^n \times a^p = a^{m+n+p}\)

• Multiple terms: Add all exponents when base is the same

• Same base: Rule only applies when all bases are identical

Rules and methods, laws,...
\( a^m \times a^n = a^{m+n} \)
Product Rule for Exponents
Basic Product
\( a^m \times a^n = a^{m+n} \)
Add exponents, keep base
Multiple Terms
\( a^m \times a^n \times a^p = a^{m+n+p} \)
Add all exponents, keep base
Negative Exponent
\( a^m \times a^{-n} = a^{m-n} \)
Subtract when adding negative
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

Product rule: When multiplying powers with the same base, add the exponents

Like bases: Powers that have the same base number

Exponent addition: The process of combining exponents when multiplying

Power multiplication methods:
  1. Identify same bases: Ensure all terms have identical bases
  2. Add exponents: Sum all the exponents together
  3. Keep base: The result has the same base as the original terms
  4. Simplify: Calculate if needed or leave in exponential form
Tip 1: Only apply this rule when bases are exactly the same
Tip 2: Add exponents even when they're negative
Tip 3: Remember: \(a^0 = 1\) for any \(a \neq 0\)
Tip 4: This rule works for any number of terms with the same base
Common errors: Applying rule to different bases, forgetting to add negative exponents correctly, misidentifying same bases.
Exam preparation: Practice with various exponent combinations, memorize the product rule, watch for different bases.
Formulas to know by heart:

• Basic product rule: \(a^m \times a^n = a^{m+n}\)

• Multiple terms: \(a^m \times a^n \times a^p = a^{m+n+p}\)

• With negative exponents: \(a^m \times a^{-n} = a^{m-n}\)

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

• One exponent: \(a^1 = a\)

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

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

Original Expression
\(\left(\frac{1}{2}\right)^3 \times \left(\frac{1}{2}\right)^2\)
Same Base
\(\frac{1}{2}\)
Add Exponents
\(3 + 2 = 5\)
Result
\(\left(\frac{1}{2}\right)^5\)
Step 1: Identify the same base

Both terms have the same fractional base: \(\frac{1}{2}\)

Step 2: Apply the product rule

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

Step 3: Add the exponents

\(3 + 2 = 5\)

Step 4: Write the result

\(\left(\frac{1}{2}\right)^5\)

Step 5: Calculate if needed

\(\left(\frac{1}{2}\right)^5 = \frac{1}{32}\)

\( \left(\frac{1}{2}\right)^3 \times \left(\frac{1}{2}\right)^2 = \left(\frac{1}{2}\right)^5 = \frac{1}{32} \)
Final answer:

\( \left(\frac{1}{2}\right)^3 \times \left(\frac{1}{2}\right)^2 = \left(\frac{1}{2}\right)^5 \)

Applied rules:

• Product rule: Applies to fractional bases too

• Same base: Both terms have base \(\frac{1}{2}\)

• Exponent addition: Add exponents when multiplying

5 Variable Base
Exercise 5
Calculate: \( x^4 \times x^7 \) where \(x \neq 0\)
Definition:

Variable base: The product rule works with variables too: \(x^m \times x^n = x^{m+n}\)

Original Expression
\(x^4 \times x^7\)
Same Base
\(x\)
Add Exponents
\(4 + 7 = 11\)
Result
\(x^{11}\)
Step 1: Identify the same base

Both terms have the same variable base: \(x\)

Step 2: Apply the product rule

\(x^4 \times x^7 = x^{4+7}\)

Step 3: Add the exponents

\(4 + 7 = 11\)

Step 4: Write the result

\(x^{11}\)

Step 5: Note restrictions

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

\( x^4 \times x^7 = x^{11} \)
Final answer:

\( x^4 \times x^7 = x^{11} \)

Applied rules:

• Product rule: Works with variable bases

• Variable exponents: Add exponents algebraically

• General form: \(x^m \times x^n = x^{m+n}\)

Key Concepts: Laws, Methods, Rules, Definitions
\( a^m \times a^n = a^{m+n} \)
Fundamental Product 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 (the bottom number)

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

Product rule: When multiplying powers with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\)

Like bases: Powers that have identical base numbers

Exponent addition: The operation performed on exponents when multiplying like bases

Algebraic expression: Using variables as bases in power expressions

Complete multiplication methodology:
  1. Identify like bases: Check that all terms have identical bases
  2. Verify conditions: Ensure bases are exactly the same (including signs)
  3. Apply product rule: Add all exponents together
  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: Add exponents even when they're negative or fractional
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, forgetting to add negative exponents correctly, not recognizing same bases.
Exam preparation: Practice with various base types, memorize the rule, watch for disguised same bases.
Formulas to know by heart:

• Basic product rule: \(a^m \times a^n = a^{m+n}\)

• Multiple terms: \(a^m \times a^n \times a^p = a^{m+n+p}\)

• With negative exponents: \(a^m \times a^{-n} = a^{m-n}\)

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

• Variable base: \(x^m \times x^n = x^{m+n}\)

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

Exercise with Visualization: Power Multiplication Patterns
Exercise 6: Multiplication of Powers with Same Base
Observe these multiplication patterns:
\( 2^1 \times 2^2 = 2^3 \)
\( 2^2 \times 2^3 = 2^5 \)
\( 2^3 \times 2^4 = 2^7 \)
\( 2^4 \times 2^5 = 2^9 \)

Analysis: The chart shows how exponents add when multiplying powers with the same base.

  • \( 2^1 \times 2^2 = 2^{1+2} = 2^3 \)
  • \( 2^2 \times 2^3 = 2^{2+3} = 2^5 \)
  • \( 2^3 \times 2^4 = 2^{3+4} = 2^7 \)
  • \( 2^4 \times 2^5 = 2^{4+5} = 2^9 \)

Questions & Answers

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

Answer: This makes perfect sense when you think about what exponents mean!

Let's look at \(2^3 \times 2^4\):

  • \(2^3 = 2 \times 2 \times 2\) (three 2's)
  • \(2^4 = 2 \times 2 \times 2 \times 2\) (four 2's)
  • So \(2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2)\)
  • This gives us seven 2's multiplied together: \(2^7\)

Since we started with 3 twos and added 4 more twos, we end up with \(3 + 4 = 7\) twos total.

That's why the rule is: add the exponents when multiplying like bases!

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

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

Examples:

  • \(2^3 \times 2^4 = 2^{3+4} = 2^7\) ✓ (same base)
  • \(2^3 \times 3^4 \neq (2 \times 3)^{3+4}\) ✗ (different bases)
  • \(2^3 \times 3^4 = 8 \times 81 = 648\) (calculate separately)

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

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

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

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

  1. Check same base: Ensure all terms have identical bases
  2. Verify exponent addition: Make sure you added the exponents correctly
  3. Expand to check: For small exponents, expand and multiply manually
  4. Estimation: Check if your answer is reasonable

Example verification: For \(2^3 \times 2^4 = 2^7\)

  • Same base check: Both have base 2 ✓
  • Exponent addition: \(3 + 4 = 7\) ✓
  • Manual check: \(8 \times 16 = 128\) and \(2^7 = 128\) ✓

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