Topic 1.1 · Unit 1 · CA CCSS N-RN.1–2, 8-EE.1
Exponent rules and scientific notation
Every exponent rule comes from one fact: an exponent counts how many copies are multiplied. If you forget a rule, expand a small case and read the answer off; that takes ten seconds and never misleads you.
- Multiplying, same base: add the exponents. \( x^3 \cdot x^4 = x^7 \), because three copies times four copies is seven copies.
- Dividing, same base: subtract. \( \frac{x^7}{x^4} = x^3 \).
- Power of a power: multiply. \( (x^3)^4 = x^{12} \), four groups of three copies.
- Power of a product: the exponent reaches everything inside. \( (2x)^3 = 8x^3 \), not \( 2x^3 \). The 2 is cubed as well.
- Zero and negative: \( x^0 = 1 \) for any \( x \neq 0 \), and \( x^{-n} = \frac{1}{x^n} \). A negative exponent means reciprocal, never a negative answer.
- Scientific notation: \( a \times 10^n \) with \( 1 \le a < 10 \). Multiply by adding exponents and multiplying the fronts; if the front leaves that range, fix it by shifting one place.
Where marks are lost: \( (-3)^2 = 9 \) but \( -3^2 = -9 \), without brackets the exponent binds tighter than the minus. And \( 2^{-3} = \frac{1}{8} \), a positive number; students frequently write \( -8 \).
Example 1: combining rules. Simplify \( \frac{(2x^3)^2 \cdot x^4}{4x^5} \).
Handle the bracket first: \( (2x^3)^2 = 4x^6 \), the 2 is squared too. Numerator: \( 4x^6 \cdot x^4 = 4x^{10} \). Now divide: \( \frac{4x^{10}}{4x^5} = x^5 \).
Example 2: a negative exponent. Write \( \frac{3x^{-2}}{y^{-4}} \) with positive exponents only.
A factor with a negative exponent moves across the fraction bar and the sign flips. The \( x^{-2} \) goes down, the \( y^{-4} \) comes up: \( \frac{3y^4}{x^2} \). Note the 3 does not move; it has no negative exponent.
Example 3: scientific notation. Compute \( (4 \times 10^5)(3 \times 10^{-8}) \).
Multiply the fronts and add the exponents: \( 12 \times 10^{-3} \). But 12 is not between 1 and 10, so shift one place: \( 1.2 \times 10^{-2} \). Making the front smaller means making the exponent bigger, so the value is unchanged.
1–6 apply one rule, 7–12 combine them, 13–15 use scientific notation and diagnosis.
- Simplify \( x^5 \cdot x^3 \).
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Same base, so add: \( x^8 \)
- Simplify \( \frac{y^9}{y^4} \).
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Subtract: \( y^5 \)
- Simplify \( (a^4)^3 \).
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Multiply: \( a^{12} \)
- Simplify \( (3x)^2 \).
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The exponent reaches the 3 as well: \( 9x^2 \)
- Evaluate \( 7^0 \).
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Any non-zero base to the zero power. 1
- Write \( 4^{-2} \) as a fraction.
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\( \frac{1}{4^2} = \frac{1}{16} \). Positive, not negative. \( \frac{1}{16} \)
- Simplify \( (2x^3)^4 \).
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\( 2^4 = 16 \) and \( (x^3)^4 = x^{12} \). \( 16x^{12} \)
- Simplify \( \frac{12x^7}{3x^2} \).
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Divide the numbers, subtract the exponents. \( 4x^5 \)
- Simplify \( \frac{(x^2)^5}{x^3} \).
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\( x^{10} \div x^3 \). \( x^7 \)
- Write \( \frac{5x^{-3}}{y^{-2}} \) with positive exponents.
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\( x^{-3} \) moves down, \( y^{-2} \) moves up; the 5 stays. \( \frac{5y^2}{x^3} \)
- Simplify \( (4x^2y)(3xy^3) \).
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Numbers: \( 12 \). \( x \): \( 2 + 1 = 3 \). \( y \): \( 1 + 3 = 4 \). \( 12x^3y^4 \)
- Simplify \( \left( \frac{x^3}{2} \right)^2 \).
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Square top and bottom: \( \frac{x^6}{4} \). \( \frac{x^6}{4} \)
- Compute \( (5 \times 10^4)(6 \times 10^3) \) in scientific notation.
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\( 30 \times 10^7 \); 30 is out of range, so shift: \( 3 \times 10^8 \). \( 3 \times 10^8 \)
- A student writes \( -4^2 = 16 \). Find the error.
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Without brackets the exponent applies only to the 4, and the minus is applied after: \( -(4^2) = -16 \). It would be 16 only if written \( (-4)^2 \). \( -16 \)
- A student writes \( (3x)^2 = 3x^2 \). Find the error.
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The exponent applies to everything inside the bracket, the 3 included: \( 3^2 x^2 = 9x^2 \). Check with \( x = 1 \): \( (3)^2 = 9 \), while their version gives 3. \( 9x^2 \)