IB · Unit 1 · 1A Exponents
Same base: add when you multiply, subtract when you divide, multiply when you raise a power to a power.
Product rule
Same base, multiply: add the exponents.
\(a^m \cdot a^n = a^{m+n}\)
\(x^2 \cdot x^3 = x^5\)
Quotient rule
Same base, divide: subtract the exponents.
\(\dfrac{a^m}{a^n} = a^{m-n}\)
\(\dfrac{x^5}{x^2} = x^3\)
Power rule
A power raised to a power: multiply the exponents.
\((a^m)^n = a^{mn}\)
\((x^2)^3 = x^6\)
Coefficients: commute first
Use the commutative property to put the numbers together and the matching bases together, then simplify.
\(6x^3 \cdot 2x^4 = 6 \cdot 2 \cdot x^3 \cdot x^4\)
One rule. Keep the same base.
Same base, so add the exponents.
\(x^{3+5} = x^8\)
Answer: \(x^8\)
Same base, so add the exponents.
\(y^{2+6} = y^8\)
Answer: \(y^8\)
\(a = a^1\)
Add the exponents.
\(a^{4+1} = a^5\)
Answer: \(a^5\)
Same base, so subtract the exponents.
\(m^{9-3} = m^6\)
Answer: \(m^6\)
Power of a power: multiply the exponents.
\(w^{2 \cdot 4} = w^8\)
Answer: \(w^8\)
Same base, so add the exponents.
\(4^{3+2} = 4^5\)
Answer: \(4^5\)
Use the commutative property: numbers together, matching bases together.
Use the commutative property to rearrange: numbers together, matching bases together.
\(6x^3 \cdot 2x^4 = 6 \cdot 2 \cdot x^3 \cdot x^4\)
\(6 \cdot 2 = 12\)
\(x^3 \cdot x^4 = x^{3+4} = x^7\)
\(12x^7\)
Answer: \(12x^7\)
Use the commutative property to rearrange.
\(5p^2 \cdot 3p^6 = 5 \cdot 3 \cdot p^2 \cdot p^6\)
\(5 \cdot 3 = 15\)
\(p^2 \cdot p^6 = p^{2+6} = p^8\)
\(15p^8\)
Answer: \(15p^8\)
Rearrange: numbers together, matching bases together.
\(\dfrac{12n^9}{4n^2} = \dfrac{12}{4} \cdot \dfrac{n^9}{n^2}\)
\(\dfrac{12}{4} = 3\)
\(\dfrac{n^9}{n^2} = n^{9-2} = n^7\)
\(3n^7\)
Answer: \(3n^7\)
Rearrange: numbers together, matching bases together.
\(\dfrac{8t^8}{2t^3} = \dfrac{8}{2} \cdot \dfrac{t^8}{t^3}\)
\(\dfrac{8}{2} = 4\)
\(\dfrac{t^8}{t^3} = t^{8-3} = t^5\)
\(4t^5\)
Answer: \(4t^5\)
The outer exponent hits every factor.
\((3x^2)^3 = 3^3 \cdot (x^2)^3\)
\(3^3 = 27\)
\((x^2)^3 = x^{2 \cdot 3} = x^6\)
\(27x^6\)
Answer: \(27x^6\)
Use the commutative property to rearrange.
\(2a^3 b \cdot 5a b^4 = 2 \cdot 5 \cdot a^3 \cdot a \cdot b \cdot b^4\)
\(2 \cdot 5 = 10\)
\(a^3 \cdot a = a^{3+1} = a^4\)
\(b \cdot b^4 = b^{1+4} = b^5\)
\(10a^4 b^5\)
Answer: \(10a^4 b^5\)
Use more than one rule. Leave the answer with positive exponents.
\((x^2)^3 = x^{2 \cdot 3} = x^6\)
Use the commutative property: matching bases together.
\(\dfrac{x^6 \cdot x^4}{x^5} = \dfrac{x^{6+4}}{x^5} = x^{10-5} = x^5\)
Answer: \(x^5\)
\((2y^3)^2 = 2^2 \cdot (y^3)^2 = 4y^6\)
Use the commutative property to rearrange.
\(4y^6 \cdot 3y = 4 \cdot 3 \cdot y^6 \cdot y\)
\(4 \cdot 3 = 12\)
\(y^6 \cdot y = y^{6+1} = y^7\)
\(12y^7\)
Answer: \(12y^7\)
Rearrange: numbers together, matching bases together.
\(\dfrac{18a^5 b^3}{6a^2 b} = \dfrac{18}{6} \cdot \dfrac{a^5}{a^2} \cdot \dfrac{b^3}{b}\)
\(\dfrac{18}{6} = 3\)
\(\dfrac{a^5}{a^2} = a^{5-2} = a^3\)
\(\dfrac{b^3}{b} = b^{3-1} = b^2\)
\(3a^3 b^2\)
Answer: \(3a^3 b^2\)
\((m^4 n)^3 = (m^4)^3 \cdot n^3 = m^{12} n^3\)
Use the commutative property to rearrange.
\(m^{12} n^3 \cdot m n^2 = m^{12} \cdot m \cdot n^3 \cdot n^2\)
\(m^{12} \cdot m = m^{13}\)
\(n^3 \cdot n^2 = n^5\)
\(m^{13} n^5\)
Answer: \(m^{13} n^5\)
\((3x^2)^2 = 3^2 \cdot (x^2)^2 = 9x^4\)
\(\dfrac{9x^4}{x^3} = 9 \cdot \dfrac{x^4}{x^3} = 9x\)
Answer: \(9x\)
Use the commutative property to rearrange.
\(\dfrac{4 \cdot 5 \cdot x^2 \cdot x \cdot y^3 \cdot y}{2 \cdot y^2}\)
\(\dfrac{4 \cdot 5}{2} = 10\)
\(x^2 \cdot x = x^3\)
\(\dfrac{y^3 \cdot y}{y^2} = y^{3+1-2} = y^2\)
\(10x^3 y^2\)
Answer: \(10x^3 y^2\)
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