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Standard Form – Calculations
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Foundational Skills
State the index law for multiplication
\[ 10^3 \times 10^4 = 10^{?} \]
Add the indices when multiplying.
State the index law for division
\[ 10^5 \div 10^2 = 10^{?} \]
Subtract the indices when dividing.
Add indices (positive and mixed)
\[ 10^5 \times 10^{-2} = 10^{?} \]
Find the missing power when multiplying.
Subtract indices (positive result)
\[ 10^6 \div 10^2 = 10^{?} \]
Division with positive result.
Subtract indices (negative result)
\[ 10^2 \div 10^5 = 10^{?} \]
Division with negative result.
Subtract a negative index
\[ 10^3 \div 10^{-2} = 10^{?} \]
Division by a negative power.
Multiplication: No Adjustment
Multiply (positive powers)
\[ (2 \times 10^3) \times (3 \times 10^4) \]
Both positive, coefficient 1-9.
Multiply (mixed power signs)
\[ (4 \times 10^5) \times (2 \times 10^{-2}) \]
One positive, one negative power.
Multiply (both negative powers)
\[ (3 \times 10^{-2}) \times (2 \times 10^{-3}) \]
Both negative powers.
Multiply with decimal coefficient
\[ (2.5 \times 10^4) \times (2 \times 10^3) \]
Decimal coefficient, no adjustment.
Multiplication: With Adjustment
Multiply (adjustment needed)
\[ (4 \times 10^3) \times (5 \times 10^2) \]
Coefficient product exceeds 10.
Multiply (mixed powers, adjust)
\[ (6 \times 10^4) \times (3 \times 10^{-2}) \]
Mixed powers, needs adjustment.
Multiply decimals (adjustment)
\[ (2.5 \times 10^3) \times (6 \times 10^2) \]
Decimal coefficients, adjust needed.
Division: No Adjustment
Divide (both positive powers)
\[ (8 \times 10^6) \div (2 \times 10^3) \]
Both positive, clean quotient.
Divide (subtracting negative)
\[ (6 \times 10^4) \div (3 \times 10^{-2}) \]
Second power is negative.
Divide (both negative powers)
\[ (8 \times 10^{-2}) \div (4 \times 10^{-5}) \]
Both negative powers.
Divide with decimal coefficients
\[ (8.4 \times 10^5) \div (2.1 \times 10^2) \]
Decimal coefficients, clean division.
Division: With Adjustment
Divide (coefficient < 1)
\[ (3 \times 10^5) \div (6 \times 10^2) \]
Quotient less than 1, needs adjusting.
Divide negative powers (adjust)
\[ (2 \times 10^{-3}) \div (5 \times 10^{-1}) \]
Negative powers, adjustment needed.
Addition: Same Power
Add with same power (no adjust)
\[ (3.2 \times 10^4) + (4.5 \times 10^4) \]
Same power, sum < 10.
Add with same power (adjust)
\[ (6.5 \times 10^3) + (5.8 \times 10^3) \]
Same power, sum ≥ 10.
Addition: Different Powers
Add (powers differ by 1)
\[ (5 \times 10^4) + (3 \times 10^3) \]
Convert to same power first.
Add (powers differ by 2)
\[ (4 \times 10^5) + (7 \times 10^3) \]
Larger power difference.
Subtraction: Same Power
Subtract same power (no adjust)
\[ (8.5 \times 10^4) – (3.2 \times 10^4) \]
Same power, result ≥ 1.
Subtract same power (adjust)
\[ (3.2 \times 10^5) – (2.8 \times 10^5) \]
Same power, result < 1.
Subtraction: Different Powers
Subtract different powers
\[ (6 \times 10^5) – (4 \times 10^4) \]
Convert to same power first.
Subtract with decimal coefficient
\[ (4.3 \times 10^5) – (2 \times 10^4) \]
First number has 1 d.p.
Powers of Standard Form
Square (no adjustment)
\[ (3 \times 10^4)^2 \]
Squared coefficient < 10.
Square (adjustment needed)
\[ (4 \times 10^3)^2 \]
Squared coefficient ≥ 10.
Cube a standard form number
\[ (2 \times 10^2)^3 \]
Cube coefficient, triple power.
Calculator Skills
Calculator entry and calculation
\[ \text{Use EXP/×10}^x \text{ button} \]
Enter and calculate standard form.
Interpret calculator display
\[ \text{2.5E-04} = \text{?} \]
Convert E notation to standard form.
Special Cases
Multiply by a power of 10
\[ (3.5 \times 10^4) \times 10^3 \]
Just add to the power.
Divide by a power of 10
\[ (6.2 \times 10^5) \div 10^2 \]
Just subtract from the power.
Contextual multiplication
\[ \text{Speed} \times \text{Time} \]
Real-world multiplication context.
Contextual division
\[ \text{Total} \div \text{Number} \]
Real-world division context.
Multi-step calculation
\[ \frac{(a \times 10^m)(b \times 10^n)}{c \times 10^p} \]
Multiply and divide combined.
Timer (Optional)
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