Powers and Notation Task Cards
Module 1 Test Review • Exponent Rules & Scientific Notation
Set 1 of 7
CARD 01 Scientific Form
Which expression is written in correct scientific notation?
A. \(0.45 \times 10^5\)
B. \(4.5 \times 10^5\)
C. \(45 \times 10^4\)
D. \(450 \times 10^3\)
Rule: Scientific Notation Form
Scientific notation must be written as \(a \times 10^n\), where the coefficient \(a\) is at least 1 and less than 10 (\(1 \le |a| < 10\)), and \(n\) is an integer.
CARD 02 Comparing Values
Compare the numbers written in scientific notation:
\(2.4 \times 10^6\) \(1.05 \times 10^9\)
Greater than (>) Less than (<) Equal to (=)
Rule: Comparing Scientific Powers
Compare powers of 10 first. A larger positive exponent always yields a larger number. If exponents match, compare the leading coefficients.
CARD 03 Standard vs Scientific
Compare the standard number and scientific expression:
\(2,500\) \(2.5 \times 10^3\)
Greater than (>) Less than (<) Equal to (=)
Rule: Converting for Comparison
To compare numbers in mixed formats, convert both to the same format: \(2.5 \times 10^3 = 2.5 \times 1,000 = 2,500\).
CARD 04 Product Rule
The rule for MULTIPLYING POWERS with the same base is to keep the base and:
Keep the base and the exponents.
a. add b. subtract c. multiply d. divide
Rule: Product of Powers
Formula: \(a^m \cdot a^n = a^{m+n}\). When multiplying like bases, keep the base unchanged and add the exponents.
Module 1 Test Review • Exponent Rules & Scientific Notation
Set 2 of 7
CARD 05 Quotient Rule
The rule for DIVIDING POWERS with the same base is to keep the base and:
Keep the base and the exponents.
a. add b. subtract c. multiply d. divide
Rule: Quotient of Powers
Formula: \(\frac{a^m}{a^n} = a^{m-n}\). When dividing like bases, keep the base unchanged and subtract the denominator exponent from the numerator exponent.
CARD 06 Multi-Step Powers
Which has the same value as the expression?
\[ \frac{f^9 \times f^3}{f^4} \]
A. \(f^{23}\) B. \(f^3\) C. \(f^8\) D. \(f^2\)
Rule: Order of Exponent Operations
Numerator first: \(f^9 \cdot f^3 = f^{9+3} = f^{12}\). Then divide by denominator: \(f^{12-4} = f^8\).
CARD 07 True / False
Evaluate whether the statement is true or false:
“When writing a number in scientific notation, the first number must be greater than or equal to 1 and less than 10.”
True
False
Rule: Scientific Leading Factor
In \(a \times 10^b\), the first factor \(a\) must have exactly one non-zero digit to the left of the decimal: \(1 \le |a| < 10\).
CARD 08 Negative Powers of 10
When converting from scientific notation to standard form, if the exponent of 10 is negative, the resulting number will be:
Less than one
Greater than one
Same as original factor
Zero
Rule: Negative Power Magnitude
A negative power \(10^{-n} = \frac{1}{10^n}\) shifts the decimal point left, producing a decimal strictly between 0 and 1 (less than one).
Module 1 Test Review • Exponent Rules & Scientific Notation
Set 3 of 7
CARD 09 Real-World Scientific
A grain of sand weighs \(0.000004404\text{ kg}\). How is this weight written in scientific notation?
\(4.404 \times 10^{-6}\)
\(4.404 \times 10^{-7}\)
\(4.44 \times 10^{-7}\)
\(4.44 \times 10^{-6}\)
Rule: Small Decimal to Scientific
Move decimal right until 1 non-zero digit is left of the decimal: \(0.\underbrace{000004}_{6\text{ places}}404 \implies 4.404 \times 10^{-6}\).
CARD 10 Greatest Value
Which of the following has the greatest numerical value?
A. \((3^2)^3\)
B. \(3^2 \times 3^3\)
C. \((3^{-2})^3\)
D. \(\frac{3^3}{3^2}\)
Rule: Evaluating Powers of 3
Simplify to single powers: \((3^2)^3 = 3^6 = 729\); \(3^2 \times 3^3 = 3^5 = 243\); \((3^{-2})^3 = 3^{-6}\); \(\frac{3^3}{3^2} = 3^1 = 3\).
CARD 11 Power to a Power
The rule for TAKING THE POWER OF A POWER is to keep the base and:
Keep the base and the exponents.
a. add b. subtract c. multiply d. divide
Rule: Power of a Power Property
Formula: \((a^m)^n = a^{m \cdot n}\). When raising a power to another power, keep the base and multiply the exponents.
CARD 12 Power of a Product
The rule for TAKING THE POWER OF A PRODUCT is to:
the exponent outside of the parenthesis to each term inside.
a. add b. subtract c. divide d. distribute
Rule: Power of a Product
Formula: \((a \cdot b)^n = a^n \cdot b^n\). Distribute the outer power to every factor located inside parentheses.
Module 1 Test Review • Exponent Rules & Scientific Notation
Set 4 of 7
CARD 13 Zero Exponent
Whenever you raise a (non-zero) base to the POWER OF ZERO, your answer will always be:
Base\(^0 =\)
a. negative b. 0 c. 1 d. unknown
Rule: Zero Exponent Property
Formula: \(a^0 = 1\) (where \(a \neq 0\)). Any non-zero base raised to power zero is always equal to 1.
CARD 14 Negative Exponent Rule
Evaluate whether the statement is true or false:
“In order to simplify an exponential expression with a negative exponent, you can make the exponent positive and place the expression in the denominator of a fraction with 1 as the numerator.”
a. TRUE
b. FALSE
Rule: Negative Exponent Reciprocal
Formula: \(a^{-n} = \frac{1}{a^n}\). Moving a power across the fraction bar changes the sign of its exponent from negative to positive.
CARD 15 Rewrite Product
Rewrite the expression using a single base and exponent:
\[ 6^5 \cdot 6^4 \]
Option A: \(6^9\)
Option B: \(36^{20}\)
Rule: Keep Base, Add Powers
\(6^5 \cdot 6^4 = 6^{5+4} = 6^9\). Do not multiply bases: \(6 \times 6 \neq 36\). Keep the base 6 intact!
CARD 16 Variable Product
Rewrite the expression using a base and exponent:
\[ m^3 \cdot m^2 \]
a. \(m^5\)
b. \(m^1\)
c. \(m^6\)
Rule: Variable Base Multiplication
With algebraic bases, the same product rule applies: \(m^3 \cdot m^2 = m^{3+2} = m^5\).
Module 1 Test Review • Exponent Rules & Scientific Notation
Set 5 of 7
CARD 17 Algebraic Exponents
Rewrite the expression using a single base and exponent:
\[ 2^c \cdot 2^w \]
a. \(2^{cw}\)
b. \(2^c 2^w\)
c. \(2^{c+w}\)
Rule: Variable Exponent Sum
Add variable exponents symbolically: \(2^c \cdot 2^w = 2^{c+w}\). Exponents are added together, not multiplied!
CARD 18 True Statement
Which equation represents a TRUE statement?
A. \(19^{21} \times 19^3 = 19^{63}\)
B. \((24^8)^5 = 24^{40}\)
C. \(20^1 \times 20^0 = 20^0\)
D. \(\frac{16^{40}}{16^{20}} = 16^2\)
Rule: Power of a Power Product
In \((24^8)^5\), exponents multiply: \(8 \times 5 = 40 \implies 24^{40}\). In A, \(21+3=24\) (not 63). In D, \(40-20=20\) (not 2).
CARD 19 Evaluate Expression
Find the value. Write without any exponents:
\[ (5^3)^1 = \underline{\hspace{2cm}} \]
\(5^4 = 625\)
\(5^3 = 125\)
\(3^5 = 15\)
\(3^6 = 18\)
Rule: Power of 1 & Calculation
Multiply exponents: \(3 \times 1 = 3 \implies 5^3\). Then calculate: \(5^3 = 5 \times 5 \times 5 = 125\).
CARD 20 Zero Power Product
Find the value of the expression:
\[ 3^0 \times 3^4 = \underline{\hspace{2cm}} \]
0
1
\(3^4\)
\(3^5\)
Rule: Multiplying by Power Zero
Method 1: Add exponents: \(3^{0+4} = 3^4\). Method 2: Recall \(3^0 = 1\), so \(1 \times 3^4 = 3^4\).
Module 1 Test Review • Exponent Rules & Scientific Notation
Set 6 of 7
CARD 21 Negative Exponent
Find the value of the expression:
\[ 7^{-2} = \underline{\hspace{2cm}} \]
\(7^2\)
\(1/7^2\)
\(1/14\)
\(-14\)
Rule: Negative Exponent Reciprocal
\(a^{-n} = \frac{1}{a^n}\). A negative exponent turns into a positive exponent in the denominator: \(7^{-2} = \frac{1}{7^2} = \frac{1}{49}\).
CARD 22 Compound Power
Find the value of the following expression:
\[ \frac{(8^4)^2}{8^{11}} \]
\(8^{-13} = 1/8^{13}\)
\(8^9\)
\(8^{-3} = 1/8^3\)
\(8^3\)
Rule: Power of Power then Subtract
Numerator: \((8^4)^2 = 8^{4 \times 2} = 8^8\). Divide: \(8^8 \div 8^{11} = 8^{8-11} = 8^{-3} = \frac{1}{8^3}\).
CARD 23 Variable Quotient
Which of the following has the same value?
\[ \frac{f^9 \times f^3}{f^4} \]
A. \(f^{23}\) B. \(f^3\) C. \(f^8\) D. \(f^2\)
Rule: Combine Product then Quotient
Multiply numerator terms: \(f^9 \cdot f^3 = f^{12}\). Subtract denominator power: \(f^{12-4} = f^8\).
CARD 24 Greatest Value Check
Which of the following has the greatest value?
A. \((3^2)^3\)
B. \(3^2 \times 3^3\)
C. \((3^{-2})^3\)
D. \(\frac{3^3}{3^2}\)
Rule: Highest Exponent Yields Max
With base 3 (\(> 1\)), the highest exponent gives the greatest value: \((3^2)^3 = 3^6 = 729 > 3^5 = 243\).
Module 1 Test Review • Exponent Rules & Scientific Notation
Set 7 of 7
CARD 25 Standard Form
Which has the same value as the following expression?
\[ 3.652 \times 10^{-4} \]
A. \(0.0003652\)
B. \(0.00003652\)
C. \(36,520,000\)
D. \(36,520\)
Rule: Shift Decimal Left for Negative
An exponent of \(-4\) instructs you to move the decimal point 4 places to the left: \(0.0003652\).
CARD 26 Convert to Scientific
Convert the value into scientific notation:
\[ 42,670,000 \]
A. \(4.267 \times 10^{-7}\)
B. \(42.67 \times 10^6\)
C. \(4.267 \times 10^7\)
D. \(42.67 \times 10^{-6}\)
Rule: Move Decimal Left for Big Numbers
Move decimal 7 places left so leading factor is between 1 and 10: \(4.267 \times 10^7\). Large numbers have positive exponents!
CARD 27 Small Decimal to Scientific
Convert the value into scientific notation:
\[ 0.0000008 \]
A. \(8 \times 10^{-7}\)
B. \(80 \times 10^{-6}\)
C. \(80 \times 10^6\)
D. \(8 \times 10^7\)
Rule: Move Decimal Right for Tiny Numbers
Move decimal 7 places right to reach 8. Moving right creates a negative exponent: \(8 \times 10^{-7}\).
REFERENCE CARD All Rules
Product: \(a^m \cdot a^n = a^{m+n}\)
Quotient: \(\frac{a^m}{a^n} = a^{m-n}\)
Power of Power: \((a^m)^n = a^{m \cdot n}\)
Power of Product: \((ab)^n = a^n b^n\)
Zero Exponent: \(a^0 = 1\) \((a \neq 0)\)
Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)
Scientific Notation: \(a \times 10^n\) where \(1 \le |a| < 10\) and \(n\) is an integer.
