Solved Exercises on Multiplying Powers with Same Base in Grade 8

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

Solution: Exercises 1 to 3
1 Basic Product Rule
Exercise 1
Simplify:
\(x^3 \cdot x^5\)
Definition:

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

Product Rule Method:
  1. Verify that bases are identical
  2. Add the exponents together
  3. Keep the same base
Expression
\(x^3 \cdot x^5\)
Apply Product Rule
\(x^{3+5}\)
Add Exponents
\(x^8\)
Step 1: Verify same base

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

Step 2: Apply the product rule

\(x^3 \cdot x^5 = x^{3+5}\)

Step 3: Calculate the sum of exponents

\(3 + 5 = 8\)

Step 4: Write the final answer

\(x^8\)

\(x^3 \cdot x^5 = x^8\)
Final answer:

\(x^3 \cdot x^5 = x^8\)

Applied rules:

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

• Same Base Requirement: Only applies when bases are identical

  • Exponent Addition: Add the exponents together
  • Base Preservation: Keep the same base
    • 2 Multiple Terms with Same Base
    Exercise 2
    Simplify:
    \(x^2 \cdot x^4 \cdot x^7\)
    Definition:

    Multiple Products: When multiplying multiple powers with the same base, add all the exponents

    Expression
    \(x^2 \cdot x^4 \cdot x^7\)
    Apply Product Rule
    \(x^{2+4+7}\)
    Add All Exponents
    \(x^{13}\)
    Step 1: Verify same base

    All terms have the same base: \(x\)

    Step 2: Apply the product rule to all terms

    \(x^2 \cdot x^4 \cdot x^7 = x^{2+4+7}\)

    Step 3: Calculate the sum of all exponents

    \(2 + 4 + 7 = 13\)

    Step 4: Write the final answer

    \(x^{13}\)

    \(x^2 \cdot x^4 \cdot x^7 = x^{13}\)
    Final answer:

    \(x^2 \cdot x^4 \cdot x^7 = x^{13}\)

    Applied rules:

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

    • Extension to Multiple Terms: \(a^m \cdot a^n \cdot a^p = a^{m+n+p}\)

    • Same Base Requirement: All terms must have identical bases

    3 With Coefficients
    Exercise 3
    Simplify:
    \(3x^4 \cdot 5x^2\)
    Definition:

    Products with Coefficients: Multiply coefficients separately, then apply product rule to like bases

    Expression
    \(3x^4 \cdot 5x^2\)
    Separate Coefficients and Variables
    \((3 \cdot 5)(x^4 \cdot x^2)\)
    Multiply Coefficients
    \(15(x^4 \cdot x^2)\)
    Apply Product Rule
    \(15x^6\)
    Step 1: Separate coefficients and variables

    \(3x^4 \cdot 5x^2 = (3 \cdot 5)(x^4 \cdot x^2)\)

    Step 2: Multiply the coefficients

    \(3 \cdot 5 = 15\)

    Step 3: Apply the product rule to variables

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

    Step 4: Combine the results

    \(15 \cdot x^6 = 15x^6\)

    \(3x^4 \cdot 5x^2 = 15x^6\)
    Final answer:

    \(3x^4 \cdot 5x^2 = 15x^6\)

    Applied rules:

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

    • Coefficient Multiplication: Multiply coefficients separately

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

    Multiplying Powers with Same Base Rules and Methods
    \(a^m \cdot a^n = a^{m+n}\)
    Product Rule
    Basic Product Rule
    \(a^m \cdot a^n = a^{m+n}\)
    Multiply powers with same base by adding exponents
    Multiple Products
    \(a^m \cdot a^n \cdot a^p = a^{m+n+p}\)
    Apply to any number of like bases
    With Coefficients
    \(ka^m \cdot la^n = (k \cdot l)a^{m+n}\)
    Multiply coefficients separately
    Negative Exponents
    \(a^{-m} \cdot a^n = a^{n-m}\)
    Rule applies with negative exponents
    Zero Exponent
    \(a^m \cdot a^0 = a^m\)
    Since \(a^0 = 1\)
    Fractional Exponents
    \(a^{m/n} \cdot a^{p/q} = a^{m/n + p/q}\)
    Rule extends to fractional exponents
    Key Concepts: The product rule only applies when multiplying powers with identical bases. The base remains unchanged while exponents are added.
    Important Note: If bases are different, you cannot apply the product rule directly. Each base must be handled separately.
    Tip 1: Always check that bases are identical before applying the product rule.
    Tip 2: When multiplying terms with coefficients, handle coefficients and variables separately.
    Tip 3: Remember: Product Rule = Add exponents (not multiply like with the power rule).
    Solution: Exercises 4 to 5
    4 Complex Expression with Same Base
    Exercise 4
    Simplify:
    \(2x^3 \cdot 4x^5 \cdot x^2\)
    Definition:

    Complex Products: Multiply coefficients separately, then apply product rule to like bases with identical exponents

    Original
    \(2x^3 \cdot 4x^5 \cdot x^2\)
    Separate Coefficients and Variables
    \((2 \cdot 4 \cdot 1)(x^3 \cdot x^5 \cdot x^2)\)
    Multiply Coefficients
    \(8(x^3 \cdot x^5 \cdot x^2)\)
    Apply Product Rule
    \(8x^{3+5+2}\)
    Final
    \(8x^{10}\)
    Step 1: Separate coefficients and variables

    \(2x^3 \cdot 4x^5 \cdot x^2 = (2 \cdot 4 \cdot 1)(x^3 \cdot x^5 \cdot x^2)\)

    Note: \(x^2 = 1x^2\), so the coefficient is 1

    Step 2: Multiply all coefficients

    \(2 \cdot 4 \cdot 1 = 8\)

    Step 3: Apply product rule to variables

    \(x^3 \cdot x^5 \cdot x^2 = x^{3+5+2} = x^{10}\)

    Step 4: Combine the results

    \(8 \cdot x^{10} = 8x^{10}\)

    \(2x^3 \cdot 4x^5 \cdot x^2 = 8x^{10}\)
    Final answer:

    \(2x^3 \cdot 4x^5 \cdot x^2 = 8x^{10}\)

    Applied rules:

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

    • Coefficient Multiplication: Multiply coefficients separately

    • Extension to Multiple Terms: Add all exponents for same base

    5 Scientific Notation with Same Base
    Exercise 5
    Multiply and express in scientific notation:
    \((3 \times 10^4) \cdot (2 \times 10^3)\)
    Definition:

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

    Original
    \((3 \times 10^4) \cdot (2 \times 10^3)\)
    Rearrange Factors
    \((3 \cdot 2)(10^4 \cdot 10^3)\)
    Multiply Coefficients
    \(6(10^4 \cdot 10^3)\)
    Apply Product Rule
    \(6 \times 10^7\)
    Step 1: Rearrange using commutative property

    \((3 \times 10^4) \cdot (2 \times 10^3) = (3 \cdot 2)(10^4 \cdot 10^3)\)

    Step 2: Multiply the coefficients

    \(3 \cdot 2 = 6\)

    Step 3: Apply product rule to powers of 10

    \(10^4 \cdot 10^3 = 10^{4+3} = 10^7\)

    Step 4: Combine results

    \(6 \times 10^7\)

    \((3 \times 10^4) \cdot (2 \times 10^3) = 6 \times 10^7\)
    Final answer:

    \((3 \times 10^4) \cdot (2 \times 10^3) = 6 \times 10^7\)

    Applied rules:

    • Commutative Property: Rearrange factors

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

    • Scientific Notation: Express result properly

    Complete Guide: Multiplying Powers with Same Base, Rules, Methods, and Applications
    \(a^m \cdot a^n = a^{m+n}\)
    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

    Product Rule: The rule for multiplying powers with the same base

    Coefficient: The numerical factor in front of a variable term

    Complete methodology:
    1. Identify the structure: Confirm that all terms have the same base
    2. Separate coefficients: Handle numerical coefficients separately from variables
    3. Apply the product rule: Add the exponents while keeping the base
    4. Perform arithmetic: Calculate sums of exponents and products of coefficients
    5. Simplify: Combine the results into a single expression
    Tip 1: The product rule is \(a^m \cdot a^n = a^{m+n}\) - add exponents, don't multiply them!
    Tip 2: When multiplying terms, always handle coefficients and variables separately.
    Tip 3: For multiple terms with same base, add all exponents: \(a^m \cdot a^n \cdot a^p = a^{m+n+p}\).
    Tip 4: Always verify that bases are identical before applying the product rule.
    Common errors: Adding bases instead of exponents, applying the rule to different bases, forgetting to handle coefficients separately.
    Exam preparation: Practice with multiple terms, master coefficient handling, understand when the rule applies and when it doesn't.
    Essential rules to memorize:

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

    • Multiple Terms: \(a^m \cdot a^n \cdot a^p = a^{m+n+p}\)

    • With Coefficients: \(ka^m \cdot la^n = (kl)a^{m+n}\)

    • Power Rule: \((a^m)^n = a^{mn}\)

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

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

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

    Exercise with Visualization: Multiplying Powers Functions
    Exercise 6: Multiplying Powers Behavior
    Consider the following functions:
    \(f_1(x) = x^2 \cdot x^3 = x^5\)
    \(f_2(x) = x^4 \cdot x^1 = x^5\)
    \(f_3(x) = x^3 \cdot x^2 \cdot x^1 = x^6\)

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

    • \(f_1(x) = x^2 \cdot x^3 = x^5\) (Product rule: add exponents)
    • \(f_2(x) = x^4 \cdot x^1 = x^5\) (Commutative property of addition)
    • \(f_3(x) = x^3 \cdot x^2 \cdot x^1 = x^6\) (Multiple terms)

    Questions & Answers

    Question: Why do we add the exponents when multiplying powers with the same base? Why don't we multiply them?

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

    \(x^2 \cdot x^3\) means:

    \((x \cdot x) \cdot (x \cdot x \cdot x) = x \cdot x \cdot x \cdot x \cdot x = x^5\)

    We're counting how many times we multiply the base by itself. We have 2 x's in the first term and 3 x's in the second term, so we have a total of \(2 + 3 = 5\) x's multiplied together.

    The product rule represents repeated multiplication, which adds to the total count of how many times the base appears in the multiplication.

    Multiplying the exponents would be incorrect because it doesn't represent the actual multiplication of the terms.

    Question: What happens if I have different bases? Can I still add the exponents? Like in \(x^3 \cdot y^4\)?

    Answer: No, you cannot apply the product rule when bases are different!

    For \(x^3 \cdot y^4\):

    This expression remains as \(x^3 \cdot y^4\) or \(x^3y^4\). You cannot combine these terms because they have different bases.

    The product rule only applies when the bases are identical:

    • Same base: \(x^3 \cdot x^4 = x^{3+4} = x^7\) ✓
    • Different bases: \(x^3 \cdot y^4 = x^3y^4\) (cannot be simplified further)

    Think of it like trying to add apples and oranges - they're fundamentally different things, so you can't combine them into a single term.

    Always check that bases match before applying the product rule!

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

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

    Examples in science:

    • Area calculations: \((3 \times 10^3) \times (2 \times 10^4) = 6 \times 10^7\)
    • Volume calculations: \((1 \times 10^2)^3 = 1 \times 10^6\)
    • Physics equations: \(F = ma\) often involves powers of 10

    For \((4 \times 10^5) \cdot (3 \times 10^2)\):

    1. Separate coefficients and powers: \((4 \cdot 3)(10^5 \cdot 10^2)\)
    2. Multiply coefficients: \(12\)
    3. Apply product rule to powers: \(10^5 \cdot 10^2 = 10^7\)
    4. Result: \(12 \times 10^7\), which becomes \(1.2 \times 10^8\) in proper scientific notation

    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 multiplication? Like \(x^3 \cdot x^{-5}\)?

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

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

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

    Key points:

    • The product rule \(a^m \cdot a^n = a^{m+n}\) applies to negative exponents
    • Follow the rules for adding signed numbers
    • Positive + Negative = Sign depends on which is larger
    • Negative + Negative = More negative

    So \(x^3 \cdot x^{-5} = x^{-2}\) while \(x^{-3} \cdot x^{-5} = x^{-8}\)

    Question: How do I know when to use the product 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:

    • Product Rule: \(a^m \cdot a^n\) - Same base multiplied together
    • Quotient Rule: \(\frac{a^m}{a^n}\) - Same base divided
    • 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 multiplication with same base: Use product rule (add exponents)
    2. If you see division with same base: Use quotient rule (subtract exponents)
    3. If you see a power raised to another power: Use power rule (multiply exponents)

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

    • First: Apply product rule to multiplication: \(x^2 \cdot x^3 = x^5\)
    • Second: Apply quotient rule: \(\frac{x^5}{x^2} = x^3\)
    • Third: Apply product rule again: \(x^5 \cdot x^3 = x^8\)

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