Exponent rules — Digital SAT Math practice
Product, quotient, power and rational exponent rules.
Key terms
- Base: The number being multiplied repeatedly in an exponential expression.
- Exponent: The small number indicating how many times the base is used as a factor.
- Zero Power Property: Any nonzero base raised to the zero power equals one.
- Negative Exponent: Indicates the reciprocal of the base raised to the positive exponent.
Common traps
- Adding exponents when multiplying different bases instead of the same base.
- Forgetting to apply the outer exponent to every factor inside parentheses.
- Using x^0 = 1 when x = 0; the zero-power rule needs a nonzero base.
Key ideas to know
Simplifying Exponent Products and Quotients
Simplify the expression x5 * x3 / x2 using exponent rules. What is the final simplified form?
Combine exponents by adding for multiplication and subtracting for division: 5 + 3 - 2 = 6, so the result is x6.
Watch out: Adding exponents when dividing or subtracting when multiplying reverses the correct rule.
Power and Negative Exponent Rules
What happens to the exponent when you raise a power to another power, like (x3)4? How do you rewrite x-2 without a negative exponent?
Multiply the exponents for the power rule: (x3)4 = x3*4 = x12. For negative exponents, flip the base to make it positive: x-2 = 1/x2.
Watch out: Adding exponents instead of multiplying them, or forgetting to flip the fraction for negative exponents.
Zero exponent
A student claims that 50 + 23 equals 8 because 50 is 0. What is wrong with this step, and what is the correct sum?
Any nonzero base raised to the zero power equals 1, not 0. The correct calculation is 1 + 8 = 9.
Watch out: Treating the zero exponent like a zero multiplier instead of a rule that gives 1 for any nonzero base.
Solving Exponential Equations with Matching Bases
To solve 32x+1 = 37, what is the most efficient next step, and what value does x equal?
Set the exponents equal to each other because the bases are identical: 2x + 1 = 7. Solve for x to get x = 3.
Watch out: Trying to expand or calculate the actual values of the exponential terms instead of equating the exponents directly.
Sample questions
Question 1
Which expression is equivalent to (3x²y³)²?
- 3x⁴y⁶
- 6x⁴y⁶
- 9x⁴y⁶
- 9x⁴y⁵
Show answer and explanation
Answer: 9x⁴y⁶
Square each factor: 3² = 9, (x²)² = x⁴, (y³)² = y⁶. Result: 9x⁴y⁶.
Question 2
Which expression is equivalent to x⁵ᐟ² · x³ᐟ² for x > 0?
- x⁴
- x¹⁵ᐟ⁴
- x⁸
- x¹⁵ᐟ²
Show answer and explanation
Answer: x⁴
When multiplying powers with the same base, add exponents: 5/2 + 3/2 = 4, so the product is x⁴.