Study guide
3.2 Scientific Notation & Operations
ACCUPLACER Quantitative Reasoning, Algebra, and Statistics Exam
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1 Chapter 1: Rational Numbers & Operations
1.1 Rational Number Arithmetic & Signed Numbers 1.2 Fractions, Decimals, & Mixed Numbers Operations 1.3 Absolute Value, Number Line, & Estimation
2 Chapter 2: Ratios, Proportions & Rates
2.1 Ratios, Unit Rates, & Dimensional Analysis 2.2 Proportions & Direct/Inverse Variation 2.3 Percent Applications, Markup, Discount, & Simple Interest
3 Chapter 3: Exponents & Scientific Notation
3.1 Integer Exponents & Laws of Exponents 3.2 Scientific Notation & Operations
4 Chapter 4: Algebraic Expressions
4.1 Evaluating & Translating Algebraic Expressions 4.2 Simplifying, Combining Like Terms, & Factoring
5 Chapter 5: Linear Equations & Inequalities
5.1 Solving Single-Variable Linear Equations 5.2 Solving & Graphing Linear Inequalities 5.3 Systems of Linear Equations
6 Chapter 6: Linear Applications & Graphing
6.1 Slope & Rate of Change 6.2 Forms of Linear Equations & Graphing 6.3 Linear Modeling & Inequality Graphs
7 Chapter 7: Probability & Set Theory
7.1 Fundamental Probability Concepts 7.2 Set Theory, Operations, & Venn Diagrams
8 Chapter 8: Descriptive Statistics & Data Displays
8.1 Measures of Center: Mean, Median, Mode, & Weighted Average 8.2 Measures of Spread: Range, IQR, Variance, & Standard Deviation 8.3 Data Displays: Histograms, Box Plots, & Scatter Plots
9 Chapter 9: Geometry Concepts & Measurement
9.1 2D Geometry: Perimeter, Area, & Circles 9.2 3D Geometry: Surface Area & Volume of Solids 9.3 Triangles, Angles, & the Pythagorean Theorem
3.2 Scientific Notation & Operations
Key Takeaways Explain This Topic AI
Scientific notation expresses numbers as a × 10^n, where the coefficient satisfies 1 ≤ |a| < 10 and n is an integer.
Positive exponents (n > 0) represent large numbers (magnitude ≥ 10), while negative exponents (n < 0) represent small decimal numbers (magnitude < 1).
Multiplication in scientific notation multiplies coefficients and adds exponents: (a × 10^m)(b × 10^n) = (ab) × 10^(m+n), followed by coefficient re-normalization if ab ≥ 10.
Division in scientific notation divides coefficients and subtracts exponents: (a × 10^m) / (b × 10^n) = (a/b) × 10^(m-n), followed by re-normalization if a/b < 1.
Addition and subtraction require adjusting expressions to have identical powers of 10 before combining their coefficients.
Last updated: August 2026
Understanding Scientific Notation
In mathematics, physical sciences, engineering, and data analysis, calculations frequently involve quantities that are extraordinarily vast (such as galactic distances and national debt) or exceedingly minute (such as atomic radii and subatomic masses). Writing and calculating with numbers containing dozens of leading or trailing zeros is inefficient and prone to transcription errors. Scientific notation provides a standardized, compact system for expressing any non-zero real number using base-10 powers.
The Standard Mathematical Definition
A number is in standard scientific notation if and only if it is written in the exact form: a × 10 n a \times 10^n a × 1 0 n
where:
The Coefficient (Significand / Mantissa) a a a : A real number whose absolute value is greater than or equal to 1 1 1 and strictly less than 10 10 10 : 1 ≤ ∣ a ∣ < 10 1 \le |a| < 10 1 ≤ ∣ a ∣ < 10 This means there must be exactly one non-zero digit to the left of the decimal point .
The Base : Always 10 10 10 .
The Exponent (Order of Magnitude) n n n : An integer ( n ∈ Z = { … , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … } n \in \mathbb{Z} = \{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots \} n ∈ Z = { … , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … } ).
Testing Scientific Notation Validity
Expression Valid Scientific Notation? Reason / Correction
4.82 × 10 7 4.82 \times 10^7 4.82 × 1 0 7 Yes 1 ≤ 4.82 < 10 1 \le 4.82 < 10 1 ≤ 4.82 < 10 and 7 7 7 is an integer
− 7.05 × 10 − 4 -7.05 \times 10^{-4} − 7.05 × 1 0 − 4 Yes ∣ − 7.05 ∣ = 7.05 \vert-7.05\vert = 7.05 ∣ − 7.05∣ = 7.05 , which satisfies 1 ≤ 7.05 < 10 1 \le 7.05 < 10 1 ≤ 7.05 < 10
0.34 × 10 5 0.34 \times 10^5 0.34 × 1 0 5 No Coefficient 0.34 < 1 0.34 < 1 0.34 < 1 ; correctly written as 3.4 × 10 4 3.4 \times 10^4 3.4 × 1 0 4
52.8 × 10 3 52.8 \times 10^3 52.8 × 1 0 3 No Coefficient 52.8 ≥ 10 52.8 \ge 10 52.8 ≥ 10 ; correctly written as 5.28 × 10 4 5.28 \times 10^4 5.28 × 1 0 4
- 6.0 × 10 2.5 6.0 \times 10^{2.5} 6.0 × 1 0 2.5 No Exponent 2.5 2.5 2.5 is not an integer
- 8.1 ⋅ 5 4 8.1 \cdot 5^4 8.1 ⋅ 5 4 No Base is 5 5 5 , but scientific notation requires base 10 10 10
- Converting Between Decimal Notation and Scientific Notation
Converting between standard decimal form and scientific notation relies on tracking how many positions the decimal point moves, which corresponds directly to multiplying or dividing by powers of 10 10 10 .
1. Converting Standard Decimal Form to Scientific Notation
For Large Numbers ( ∣ N ∣ ≥ 10 ⟹ n > 0 |N| \ge 10 \implies n > 0 ∣ N ∣ ≥ 10 ⟹ n > 0 ):
Place the decimal point immediately after the first non-zero digit to create a coefficient a a a such that 1 ≤ ∣ a ∣ < 10 1 \le |a| < 10 1 ≤ ∣ a ∣ < 10 .
Count the number of places ( k k k ) the decimal point moved to the left from its original position.
The exponent is positive: n = + k n = +k n = + k .
Example : Convert 584 , 000 , 000 584,000,000 584 , 000 , 000 to scientific notation:
Place decimal after 5 5 5 : 5.84000000 ⟹ a = 5.84 5.84000000 \implies a = 5.84 5.84000000 ⟹ a = 5.84 .
The decimal moved 8 8 8 places to the left ⟹ n = + 8 \implies n = +8 ⟹ n = + 8 .
Result: 5.84 × 10 8 5.84 \times 10^8 5.84 × 1 0 8 .
For Small Decimal Numbers ( 0 < ∣ N ∣ < 1 ⟹ n < 0 0 < |N| < 1 \implies n < 0 0 < ∣ N ∣ < 1 ⟹ n < 0 ):
Place the decimal point immediately after the first non-zero digit to form a a a .
Count the number of places ( k k k ) the decimal point moved to the right from its original position.
The exponent is negative: n = − k n = -k n = − k .
Example : Convert 0.0000732 0.0000732 0.0000732 to scientific notation:
Place decimal after 7 7 7 : 7.32 ⟹ a = 7.32 7.32 \implies a = 7.32 7.32 ⟹ a = 7.32 .
The decimal moved 5 5 5 places to the right ⟹ n = − 5 \implies n = -5 ⟹ n = − 5 .
Result: 7.32 × 10 − 5 7.32 \times 10^{-5} 7.32 × 1 0 − 5 .
For Numbers Already Between 1 and 10 ( 1 ≤ ∣ N ∣ < 10 ⟹ n = 0 1 \le |N| < 10 \implies n = 0 1 ≤ ∣ N ∣ < 10 ⟹ n = 0 ):
6.19 = 6.19 × 10 0 6.19 = 6.19 \times 10^0 6.19 = 6.19 × 1 0 0 (since 10 0 = 1 10^0 = 1 1 0 0 = 1 ).
2. Converting Scientific Notation to Standard Decimal Form
If n > 0 n > 0 n > 0 (Positive Exponent) : Move the decimal point n n n places to the right , adding trailing zeros as necessary (making the number larger). 3.45 × 10 6 = 3 , 450 , 000 3.45 \times 10^6 = 3,450,000 3.45 × 1 0 6 = 3 , 450 , 000
If n < 0 n < 0 n < 0 (Negative Exponent) : Move the decimal point ∣ n ∣ |n| ∣ n ∣ places to the left , adding leading zeros after the decimal point as necessary (making the number smaller). 6.18 × 10 − 4 = 0.000618 6.18 \times 10^{-4} = 0.000618 6.18 × 1 0 − 4 = 0.000618
Reference Table: Powers of 10 and Metric Scale Prefixes
Power of 10 Standard Value Metric Prefix Symbol Common Practical Example
10 12 10^{12} 1 0 12 1 , 000 , 000 , 000 , 000 1,000,000,000,000 1 , 000 , 000 , 000 , 000 Tera- T Terabyte (data storage)
10 9 10^9 1 0 9 1 , 000 , 000 , 000 1,000,000,000 1 , 000 , 000 , 000 Giga- G Gigahertz (processor speed), national populations
- 10 6 10^6 1 0 6 1 , 000 , 000 1,000,000 1 , 000 , 000 Mega- M Megawatt (power grid output)
- 10 3 10^3 1 0 3 1 , 000 1,000 1 , 000 Kilo- k Kilometer, kilogram
- 10 0 10^0 1 0 0 1 1 1 (base unit) — Meter, gram, second
- 10 − 3 10^{-3} 1 0 − 3 0.001 0.001 0.001 Milli- m Millimeter (thickness of credit card)
- 10 − 6 10^{-6} 1 0 − 6 0.000001 0.000001 0.000001 Micro- μ \mu μ Micrometer (bacterial cell size)
- 10 − 9 10^{-9} 1 0 − 9 0.000000001 0.000000001 0.000000001 Nano- n Nanometer (semiconductor transistor gate)
- 10 − 12 10^{-12} 1 0 − 12 0.000000000001 0.000000000001 0.000000000001 Pico- p Picometer (atomic radius)
- Arithmetic Operations in Scientific Notation
Performing arithmetic with numbers in scientific notation combines standard coefficient arithmetic with the laws of exponents. Each operation requires specific techniques for managing exponents and re-normalizing the final coefficient.
1. Multiplication in Scientific Notation
To multiply two numbers in scientific notation:
Multiply the numerical coefficients: a new = a 1 ⋅ a 2 a_{\text{new}} = a_1 \cdot a_2 a new = a 1 ⋅ a 2 .
Add the exponents of 10 10 10 using the Product of Powers Rule: n new = m + n n_{\text{new}} = m + n n new = m + n .
Re-normalize if a new ≥ 10 a_{\text{new}} \ge 10 a new ≥ 10 by shifting the decimal left 1 1 1 place and increasing the exponent by + 1 +1 + 1 : ( a 1 × 10 m ) × ( a 2 × 10 n ) = ( a 1 ⋅ a 2 ) × 10 m + n (a_1 \times 10^m) \times (a_2 \times 10^n) = (a_1 \cdot a_2) \times 10^{m+n} ( a 1 × 1 0 m ) × ( a 2 × 1 0 n ) = ( a 1 ⋅ a 2 ) × 1 0 m + n
Worked Example : ( 4.5 × 10 6 ) × ( 8.0 × 10 7 ) (4.5 \times 10^6) \times (8.0 \times 10^7) ( 4.5 × 1 0 6 ) × ( 8.0 × 1 0 7 )
- Multiply coefficients: 4.5 × 8.0 = 36.0 4.5 \times 8.0 = 36.0 4.5 × 8.0 = 36.0
- Add exponents: 10 6 + 7 = 10 13 10^{6 + 7} = 10^{13} 1 0 6 + 7 = 1 0 13
- Initial combined form: 36.0 × 10 13 36.0 \times 10^{13} 36.0 × 1 0 13
Re-normalize : Since 36.0 ≥ 10 36.0 \ge 10 36.0 ≥ 10 , write 36.0 = 3.6 × 10 1 36.0 = 3.6 \times 10^1 36.0 = 3.6 × 1 0 1 : ( 3.6 × 10 1 ) × 10 13 = 3.6 × 10 14 (3.6 \times 10^1) \times 10^{13} = 3.6 \times 10^{14} ( 3.6 × 1 0 1 ) × 1 0 13 = 3.6 × 1 0 14
2. Division in Scientific Notation
To divide two numbers in scientific notation:
Divide the numerator coefficient by the denominator coefficient: a new = a 1 a 2 a_{\text{new}} = \frac{a_1}{a_2} a new = a 2 a 1 .
Subtract the denominator exponent from the numerator exponent using the Quotient of Powers Rule: n new = m − n n_{\text{new}} = m - n n new = m − n .
Re-normalize if a new < 1 a_{\text{new}} < 1 a new < 1 by shifting the decimal right 1 1 1 place and decreasing the exponent by − 1 -1 − 1 : a 1 × 10 m a 2 × 10 n = ( a 1 a 2 ) × 10 m − n \frac{a_1 \times 10^m}{a_2 \times 10^n} = \left(\frac{a_1}{a_2}\right) \times 10^{m-n} a 2 × 1 0 n a 1 × 1 0 m = ( a 2 a 1 ) × 1 0 m − n
Worked Example : 1.44 × 10 − 3 4.8 × 10 − 8 \frac{1.44 \times 10^{-3}}{4.8 \times 10^{-8}} 4.8 × 1 0 − 8 1.44 × 1 0 − 3
Divide coefficients: 1.44 4.8 = 0.3 \frac{1.44}{4.8} = 0.3 4.8 1.44 = 0.3
Subtract exponents: 10 − 3 − ( − 8 ) = 10 − 3 + 8 = 10 5 10^{-3 - (-8)} = 10^{-3 + 8} = 10^5 1 0 − 3 − ( − 8 ) = 1 0 − 3 + 8 = 1 0 5
Initial combined form: 0.3 × 10 5 0.3 \times 10^5 0.3 × 1 0 5
Re-normalize : Since 0.3 < 1 0.3 < 1 0.3 < 1 , write 0.3 = 3.0 × 10 − 1 0.3 = 3.0 \times 10^{-1} 0.3 = 3.0 × 1 0 − 1 : ( 3.0 × 10 − 1 ) × 10 5 = 3.0 × 10 4 (3.0 \times 10^{-1}) \times 10^5 = 3.0 \times 10^4 ( 3.0 × 1 0 − 1 ) × 1 0 5 = 3.0 × 1 0 4
3. Addition and Subtraction in Scientific Notation
Unlike multiplication and division, you cannot add or subtract coefficients directly unless both terms share the exact same power of 10 .
The 4-Step Alignment Algorithm for Addition & Subtraction:
Identify the term with the smaller exponent and rewrite it so its power of 10 matches the larger exponent (shift its coefficient's decimal point to the left by the difference in powers).
Factor out the common power of 10.
Add or subtract the aligned coefficients.
Re-normalize the resulting sum or difference into standard scientific form if necessary. General Form: ( a × 10 N ) + ( b × 10 N ) = ( a + b ) × 10 N \text{General Form: } (a \times 10^N) + (b \times 10^N) = (a + b) \times 10^N General Form: ( a × 1 0 N ) + ( b × 1 0 N ) = ( a + b ) × 1 0 N
Worked Example: Addition with Differing Exponents : ( 6.2 × 10 5 ) + ( 3.8 × 10 4 ) (6.2 \times 10^5) + (3.8 \times 10^4) ( 6.2 × 1 0 5 ) + ( 3.8 × 1 0 4 )
Compare exponents: 5 5 5 vs 4 4 4 . The larger exponent is 5 5 5 .
Rewrite the smaller term ( 3.8 × 10 4 3.8 \times 10^4 3.8 × 1 0 4 ) with exponent 5 5 5 : 3.8 × 10 4 = 0.38 × 10 5 3.8 \times 10^4 = 0.38 \times 10^5 3.8 × 1 0 4 = 0.38 × 1 0 5
Add coefficients: ( 6.2 + 0.38 ) × 10 5 = 6.58 × 10 5 (6.2 + 0.38) \times 10^5 = 6.58 \times 10^5 ( 6.2 + 0.38 ) × 1 0 5 = 6.58 × 1 0 5
The coefficient 6.58 6.58 6.58 satisfies 1 ≤ 6.58 < 10 1 \le 6.58 < 10 1 ≤ 6.58 < 10 , so the result is already normalized: 6.58 × 10 5 6.58 \times 10^5 6.58 × 1 0 5 .
Worked Example: Subtraction with Negative Exponents : ( 7.1 × 10 − 6 ) − ( 9.5 × 10 − 7 ) (7.1 \times 10^{-6}) - (9.5 \times 10^{-7}) ( 7.1 × 1 0 − 6 ) − ( 9.5 × 1 0 − 7 )
Compare exponents: − 6 > − 7 -6 > -7 − 6 > − 7 . The larger exponent is − 6 -6 − 6 .
Rewrite 9.5 × 10 − 7 9.5 \times 10^{-7} 9.5 × 1 0 − 7 with exponent − 6 -6 − 6 : 9.5 × 10 − 7 = 0.95 × 10 − 6 9.5 \times 10^{-7} = 0.95 \times 10^{-6} 9.5 × 1 0 − 7 = 0.95 × 1 0 − 6
Subtract coefficients: ( 7.1 − 0.95 ) × 10 − 6 = 6.15 × 10 − 6 (7.1 - 0.95) \times 10^{-6} = 6.15 \times 10^{-6} ( 7.1 − 0.95 ) × 1 0 − 6 = 6.15 × 1 0 − 6
4. Raising Scientific Notation to a Power
To raise a scientific notation expression to an integer power p p p : ( a × 10 n ) p = a p × 10 n ⋅ p (a \times 10^n)^p = a^p \times 10^{n \cdot p} ( a × 1 0 n ) p = a p × 1 0 n ⋅ p
Worked Example : ( 3.0 × 10 4 ) 3 = 3.0 3 × 10 4 ⋅ 3 = 27.0 × 10 12 = 2.7 × 10 13 (3.0 \times 10^4)^3 = 3.0^3 \times 10^{4 \cdot 3} = 27.0 \times 10^{12} = 2.7 \times 10^{13} ( 3.0 × 1 0 4 ) 3 = 3. 0 3 × 1 0 4 ⋅ 3 = 27.0 × 1 0 12 = 2.7 × 1 0 13
Real-World Applications & Applied Problems
ACCUPLACER QAS exam questions frequently contextualize scientific notation within applied word problems involving rates, distances, and scientific measurements.
Application 1: Astronomy & Light Speed Calculations
Light travels at approximately 3.0 × 10 5 km/s 3.0 \times 10^5\text{ km/s} 3.0 × 1 0 5 km/s in a vacuum. The average distance from Earth to the Sun is approximately 1.5 × 10 8 km 1.5 \times 10^8\text{ km} 1.5 × 1 0 8 km . How many seconds does it take for sunlight to travel from the Sun to Earth? Time = Distance Speed = 1.5 × 10 8 km 3.0 × 10 5 km/s \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{1.5 \times 10^8\text{ km}}{3.0 \times 10^5\text{ km/s}} Time = Speed Distance = 3.0 × 1 0 5 km/s 1.5 × 1 0 8 km Time = ( 1.5 3.0 ) × 10 8 − 5 = 0.5 × 10 3 = 500 seconds \text{Time} = \left(\frac{1.5}{3.0}\right) \times 10^{8 - 5} = 0.5 \times 10^3 = 500\text{ seconds} Time = ( 3.0 1.5 ) × 1 0 8 − 5 = 0.5 × 1 0 3 = 500 seconds 500 seconds = 8 minutes and 20 seconds 500\text{ seconds} = 8\text{ minutes and } 20\text{ seconds} 500 seconds = 8 minutes and 20 seconds
Application 2: Microbiology & Population Growth
A biological research culture contains 4.0 × 10 3 4.0 \times 10^3 4.0 × 1 0 3 bacterial cells. Under optimal laboratory conditions, the bacterial population doubles every hour for 6 6 6 hours, multiplying the original count by 2 6 = 64 2^6 = 64 2 6 = 64 . What is the final population expressed in scientific notation? Final Population = 64 × ( 4.0 × 10 3 ) = ( 64 ⋅ 4.0 ) × 10 3 = 256.0 × 10 3 \text{Final Population} = 64 \times (4.0 \times 10^3) = (64 \cdot 4.0) \times 10^3 = 256.0 \times 10^3 Final Population = 64 × ( 4.0 × 1 0 3 ) = ( 64 ⋅ 4.0 ) × 1 0 3 = 256.0 × 1 0 3 Re-normalize: 256.0 × 10 3 = ( 2.56 × 10 2 ) × 10 3 = 2.56 × 10 5 bacteria \text{Re-normalize: } 256.0 \times 10^3 = (2.56 \times 10^2) \times 10^3 = 2.56 \times 10^5\text{ bacteria} Re-normalize: 256.0 × 1 0 3 = ( 2.56 × 1 0 2 ) × 1 0 3 = 2.56 × 1 0 5 bacteria
Application 3: Computing & High-Volume Data Processing
A high-performance cloud server cluster processes 8.0 × 10 9 8.0 \times 10^9 8.0 × 1 0 9 database transactions per day. How many total transactions will the cluster process over a full leap year of 366 366 366 days? Total Transactions = 366 × ( 8.0 × 10 9 ) = ( 366 ⋅ 8.0 ) × 10 9 = 2 , 928.0 × 10 9 \text{Total Transactions} = 366 \times (8.0 \times 10^9) = (366 \cdot 8.0) \times 10^9 = 2,928.0 \times 10^9 Total Transactions = 366 × ( 8.0 × 1 0 9 ) = ( 366 ⋅ 8.0 ) × 1 0 9 = 2 , 928.0 × 1 0 9 Re-normalize: 2 , 928.0 × 10 9 = 2.928 × 10 12 transactions (2.928 trillion) \text{Re-normalize: } 2,928.0 \times 10^9 = 2.928 \times 10^{12}\text{ transactions (2.928 trillion)} Re-normalize: 2 , 928.0 × 1 0 9 = 2.928 × 1 0 12 transactions (2.928 trillion)
Common Pitfalls & ACCUPLACER Exam Traps
Leaving Answers Unnormalized : The most common error is selecting an unnormalized choice like 42 × 10 6 42 \times 10^6 42 × 1 0 6 or 0.75 × 10 − 4 0.75 \times 10^{-4} 0.75 × 1 0 − 4 . Always verify that the coefficient satisfies 1 ≤ ∣ a ∣ < 10 1 \le |a| < 10 1 ≤ ∣ a ∣ < 10 .
Adding Exponents in Addition / Subtraction : ( 2 × 10 4 ) + ( 3 × 10 4 ) = 5 × 10 4 (2 \times 10^4) + (3 \times 10^4) = 5 \times 10^4 ( 2 × 1 0 4 ) + ( 3 × 1 0 4 ) = 5 × 1 0 4 , NOT 5 × 10 8 5 \times 10^8 5 × 1 0 8 . Exponents add only during multiplication.
Sign Errors with Negative Exponent Subtraction : In 6.0 × 10 4 2.0 × 10 − 3 \frac{6.0 \times 10^4}{2.0 \times 10^{-3}} 2.0 × 1 0 − 3 6.0 × 1 0 4 , the exponent subtraction is 4 − ( − 3 ) = 4 + 3 = 7 ⟹ 3.0 × 10 7 4 - (-3) = 4 + 3 = 7 \implies 3.0 \times 10^7 4 − ( − 3 ) = 4 + 3 = 7 ⟹ 3.0 × 1 0 7 , NOT 10 1 10^1 1 0 1 .
Incorrect Direction in Decimal Shifts During Normalization :
When increasing coefficient ( 0.45 → 4.5 0.45 \to 4.5 0.45 → 4.5 ), exponent decreases ( 10 6 → 10 5 10^6 \to 10^5 1 0 6 → 1 0 5 ).
When decreasing coefficient ( 45.0 → 4.5 45.0 \to 4.5 45.0 → 4.5 ), exponent increases ( 10 6 → 10 7 10^6 \to 10^7 1 0 6 → 1 0 7 ).
Confusing Negative Numbers with Negative Exponents : In − 4.5 × 10 3 -4.5 \times 10^3 − 4.5 × 1 0 3 , the negative sign means the entire value is negative ( − 4500 -4500 − 4500 ); the positive exponent indicates a number with absolute value ≥ 10 \ge 10 ≥ 10 .