Powers of ten · Eqora guide
Calculate with scientific notation
Write very large and small numbers in standard form, then multiply, divide, add, and check them without losing a power of ten.
Scientific notation compresses a number into a coefficient and a power of ten. For example, 4,500,000 becomes 4.5 × 10^6, while 0.00032 becomes 3.2 × 10^-4. The notation is useful only when you can move between the compact form, the original value, and a sensible estimate.
The key is to treat the coefficient and the power of ten as two connected parts. The exponent records place value; it is not a decoration and it does not follow every arithmetic rule in the same way.
Normalize the coefficient
Standard scientific notation has the form a × 10^n, where 1 ≤ |a| < 10 and n is an integer. In 45 × 10^5 the coefficient is too large. Moving its decimal one place left gives 4.5, so the power must increase one step: 45 × 10^5 = 4.5 × 10^6.
For 0.072 × 10^-3, move the decimal two places right to make 7.2. Compensate by decreasing the exponent by two: 7.2 × 10^-5. Check by expanding both forms; each equals 0.000072.
Every decimal move in the coefficient needs an equal and opposite adjustment in the exponent.
Convert by tracking place value
To write 6,380,000 in scientific notation, place the decimal after the first nonzero digit: 6.38. It moved six places left, so the exponent is 6. To write 0.0000091, the decimal moves six places right, giving 9.1 × 10^-6.
A positive exponent usually represents a magnitude at least ten; a negative exponent represents a positive magnitude below one. Keep any original negative sign with the coefficient: -720,000 = -7.2 × 10^5. The exponent describes size, not whether the number is positive or negative.
Multiply and divide in two layers
For (3 × 10^4)(2.5 × 10^-2), multiply coefficients and add exponents: 7.5 × 10^2 = 750. For (8.4 × 10^7)/(2.1 × 10^3), divide coefficients and subtract exponents: 4 × 10^4 = 40,000.
Normalize only after the operation when needed. (6 × 10^5)(4 × 10^3) gives 24 × 10^8, which becomes 2.4 × 10^9. A common mistake is stopping at a coefficient of 24 or multiplying the exponents instead of adding them.
- Multiply: coefficients multiply, exponents add
- Divide: coefficients divide, exponents subtract
- Normalize the final coefficient
- Estimate the order of magnitude
Align exponents before adding
Addition and subtraction require a shared power of ten. For 3.2 × 10^5 + 4.7 × 10^4, rewrite the second term as 0.47 × 10^5. Then add coefficients: 3.67 × 10^5. Adding 3.2 + 4.7 while keeping 10^5 would incorrectly treat the terms as equal place values.
Subtraction also needs a scale check. In 6.1 × 10^-3 - 8 × 10^-4, rewrite 8 × 10^-4 as 0.8 × 10^-3, giving 5.3 × 10^-3. The result should be positive and slightly smaller than 0.0061.
Estimate the exponent first
Before exact arithmetic, predict the power of ten. Since 3.9 × 6.2 is about 24, (3.9 × 10^8)(6.2 × 10^-5) should be about 24 × 10^3, or 2.4 × 10^4 after normalization. This estimate catches a missing exponent adjustment.
On a calculator, notation may appear as E or EXP: 2.4E4 means 2.4 × 10^4, not 2.4 multiplied by an unknown E. Enter the coefficient and exponent deliberately, then compare the displayed order of magnitude with your written estimate.
Keep units and significant detail visible
Scientific notation does not change units. If a distance is 1.5 × 10^8 km and a speed is 3 × 10^5 km/s, division gives 0.5 × 10^3 s = 5 × 10^2 s. The kilometers cancel, leaving seconds; that unit check supports the operation.
Do not invent precision. If measurements are rounded, keep extra calculator digits during the work and round only according to the rule your course or problem specifies. Scientific notation shows magnitude clearly, but the number of written digits may also communicate measurement precision.
Practice, explain, and check independently
Try these without a model: convert 0.00056; normalize 31 × 10^-7; multiply (4 × 10^3)(7 × 10^-6); divide (9 × 10^8)/(3 × 10^2); and add 2.4 × 10^6 + 7 × 10^5. The results are 5.6 × 10^-4, 3.1 × 10^-6, 2.8 × 10^-2, 3 × 10^6, and 3.1 × 10^6.
If a step remains unclear, show Eqora the complete expression and your own attempt, then ask whether the issue is place value, an exponent rule, or normalization. Confirm that signs and superscripts were read correctly. Close the explanation and solve a changed example yourself; an AI tutor can misread notation or calculate incorrectly, so estimation and substitution remain your checks.
Put it into practice now
Choose one problem from your current homework or review set. Attempt it before opening Eqora, then use the app only at the point where your own reasoning stops.
- State what the problem is asking before you solve it
- Identify the first step you cannot justify
- Ask Eqora one focused follow-up about that step
- Finish with a similar problem and no solution in view
The session is complete when the method is clearer, not simply when the worksheet has one more answer.