Product • Quotient • Power • Zero & Negative • Fractional Exponents
Jump to section:
Product Rule
When multiplying powers with the same base, add the exponents.
$$a^m \cdot a^n = a^{m+n}$$
Easy example: \( x^2 \cdot x^3 = x^5 \)
Quotient Rule
When dividing powers with the same base, subtract the exponents.
$$\dfrac{a^m}{a^n} = a^{m-n}$$
Easy example: \( \dfrac{x^5}{x^2} = x^3 \)
Power Rule
When raising a power to another power, multiply the exponents.
$$(a^m)^n = a^{m \cdot n}$$
Easy example: \( (x^2)^3 = x^6 \)
Power of a Product
Distribute the outer exponent to every factor inside the parentheses.
$$(ab)^n = a^n b^n$$
Example: \( (2x)^3 = 8x^3 \)
Zero & Negative Exponents
Any non-zero number to the zero power is 1. A negative exponent means the reciprocal.
$$a^0 = 1 \quad (a \neq 0)$$
$$a^{-n} = \dfrac{1}{a^n}$$
Fractional Exponents
A fractional exponent represents a root.
$$a^{1/n} = \sqrt[n]{a}$$
$$a^{m/n} = \sqrt[n]{a^m}$$
Example: \( x^{1/2} = \sqrt{x} \)
Remember: Always write final answers with positive exponents only and combine like bases completely.
Multiply the powers. Write the answer in simplest form.
Simplify: \( x^4 \cdot x^3 \)
Add the exponents (same base):
$$x^{4+3} = x^7$$
Answer: \( x^7 \)
Simplify: \( y^2 \cdot y^5 \)
$$y^{2+5} = y^7$$
Answer: \( y^7 \)
Simplify: \( 3a^2 \cdot 4a \)
Multiply coefficients and add exponents of \( a \):
$$3 \cdot 4 \cdot a^{2+1} = 12a^3$$
Answer: \( 12a^3 \)
Simplify: \( 2b^3 \cdot 5b^4 \)
$$2 \cdot 5 \cdot b^{3+4} = 10b^7$$
Answer: \( 10b^7 \)
Simplify: \( (-3x^2)(4x^5) \)
$$(-3) \cdot 4 \cdot x^{2+5} = -12x^7$$
Answer: \( -12x^7 \)
Simplify: \( 6m^2 n \cdot 2m n^3 \)
$$6 \cdot 2 \cdot m^{2+1} \cdot n^{1+3} = 12m^3 n^4$$
Answer: \( 12m^3 n^4 \)
Simplify: \( (4x^{1/3})(3x^{2/3}) \)
$$4 \cdot 3 \cdot x^{1/3 + 2/3} = 12x^{3/3} = 12x^1 = 12x$$
Answer: \( 12x \)
Simplify: \( 2a^{3/4} \cdot 5a^{1/4} \cdot a \)
$$2 \cdot 5 \cdot a^{3/4 + 1/4 + 1} = 10a^{1 + 1} = 10a^2$$
Answer: \( 10a^2 \)
Simplify: \( (-2x^2 y)(3x y^3)(-4x) \)
Coefficients: \( (-2) \cdot 3 \cdot (-4) = 24 \)
$$x^{2+1+1} y^{1+3} = x^4 y^4$$
Answer: \( 24x^4 y^4 \)
Divide the powers. Write the answer with positive exponents only.
Simplify: \( \dfrac{x^7}{x^2} \)
$$x^{7-2} = x^5$$
Answer: \( x^5 \)
Simplify: \( \dfrac{y^9}{y^4} \)
$$y^{9-4} = y^5$$
Answer: \( y^5 \)
Simplify: \( \dfrac{12a^5}{3a^2} \)
$$\dfrac{12}{3} \cdot a^{5-2} = 4a^3$$
Answer: \( 4a^3 \)
Simplify: \( \dfrac{15b^8}{5b^3} \)
$$\dfrac{15}{5} \cdot b^{8-3} = 3b^5$$
Answer: \( 3b^5 \)
Simplify: \( \dfrac{8x^4 y^3}{2x y} \)
$$\dfrac{8}{2} \cdot x^{4-1} \cdot y^{3-1} = 4x^3 y^2$$
Answer: \( 4x^3 y^2 \)
Simplify: \( \dfrac{24m^5 n^2}{6m^2 n} \)
$$\dfrac{24}{6} \cdot m^{5-2} \cdot n^{2-1} = 4m^3 n$$
Answer: \( 4m^3 n \)
Simplify: \( \dfrac{18a^6 b^4}{6a^2 b^7} \)
$$\dfrac{18}{6} \cdot a^{6-2} \cdot b^{4-7} = 3a^4 b^{-3} = \dfrac{3a^4}{b^3}$$
Answer: \( \dfrac{3a^4}{b^3} \)
Simplify: \( \dfrac{5x^3 y}{20x y^4} \)
$$\dfrac{5}{20} \cdot x^{3-1} \cdot y^{1-4} = \dfrac{1}{4} x^2 y^{-3} = \dfrac{x^2}{4y^3}$$
Answer: \( \dfrac{x^2}{4y^3} \)
Simplify: \( \dfrac{16c^5 d^2}{4c^8 d} \)
$$\dfrac{16}{4} \cdot c^{5-8} \cdot d^{2-1} = 4c^{-3} d = \dfrac{4d}{c^3}$$
Answer: \( \dfrac{4d}{c^3} \)
Raise powers to powers and distribute outer exponents. Simplify completely.
Simplify: \( (x^3)^4 \)
$$x^{3 \cdot 4} = x^{12}$$
Answer: \( x^{12} \)
Simplify: \( (2y)^3 \)
$$2^3 \cdot y^3 = 8y^3$$
Answer: \( 8y^3 \)
Simplify: \( (a^2 b)^3 \)
$$a^{2\cdot3} b^3 = a^6 b^3$$
Answer: \( a^6 b^3 \)
Simplify: \( (3x^2)^4 \)
$$3^4 \cdot x^{2\cdot4} = 81x^8$$
Answer: \( 81x^8 \)
Simplify: \( (2a^3 b^2)^3 \)
$$2^3 \cdot a^{3\cdot3} \cdot b^{2\cdot3} = 8a^9 b^6$$
Answer: \( 8a^9 b^6 \)
Simplify: \( \left( \dfrac{x^2}{y} \right)^3 \)
$$\dfrac{x^{2\cdot3}}{y^3} = \dfrac{x^6}{y^3}$$
Answer: \( \dfrac{x^6}{y^3} \)
Simplify: \( (4x^2 y^3)^2 \cdot x \)
$$4^2 \cdot x^{2\cdot2} \cdot y^{3\cdot2} \cdot x = 16x^4 y^6 \cdot x = 16x^5 y^6$$
Answer: \( 16x^5 y^6 \)
Simplify: \( \left( \dfrac{2a^3}{b^2} \right)^3 \)
$$\dfrac{2^3 a^{3\cdot3}}{b^{2\cdot3}} = \dfrac{8a^9}{b^6}$$
Answer: \( \dfrac{8a^9}{b^6} \)
Simplify: \( (3c^2)^3 \cdot (2c)^2 \)
$$3^3 c^{6} \cdot 2^2 c^2 = 27c^6 \cdot 4c^2 = 108c^8$$
Answer: \( 108c^8 \)
Simplify. Write all answers with positive exponents only.
Simplify: \( 5^0 \)
Any non-zero number to the zero power equals 1.
Answer: \( 1 \)
Simplify: \( x^{-3} \)
$$x^{-3} = \dfrac{1}{x^3}$$
Answer: \( \dfrac{1}{x^3} \)
Simplify: \( 4^{-2} \)
$$4^{-2} = \dfrac{1}{4^2} = \dfrac{1}{16}$$
Answer: \( \dfrac{1}{16} \)
Simplify: \( 3x^0 \)
$$x^0 = 1 \), so \( 3 \cdot 1 = 3 \)
Answer: \( 3 \)
Simplify: \( \dfrac{2^{-3}}{2^{-5}} \)
$$2^{-3 – (-5)} = 2^{-3+5} = 2^2 = 4$$
Answer: \( 4 \)
Simplify: \( 5 \cdot 2^{-2} \)
$$5 \cdot \dfrac{1}{2^2} = 5 \cdot \dfrac{1}{4} = \dfrac{5}{4}$$
Answer: \( \dfrac{5}{4} \)
Simplify with no negative exponents: \( \dfrac{4 \cdot 3^{-2}}{3^{-5}} \)
$$4 \cdot 3^{-2 – (-5)} = 4 \cdot 3^{-2+5} = 4 \cdot 3^3 = 4 \cdot 27 = 108$$
Answer: \( 108 \)
Simplify: \( 6c^2 (4c^0)^3 \)
$$c^0 = 1 \), so \( (4 \cdot 1)^3 = 4^3 = 64 \)
$$6c^2 \cdot 64 = 384c^2$$
Answer: \( 384c^2 \)
Simplify with no negative exponents: \( \dfrac{5 \cdot 2^{-4}}{2^{-2}} \)
$$5 \cdot 2^{-4 – (-2)} = 5 \cdot 2^{-4+2} = 5 \cdot 2^{-2} = 5 \cdot \dfrac{1}{4} = \dfrac{5}{4}$$
Answer: \( \dfrac{5}{4} \)
Simplify expressions that contain fractional exponents. Write answers in simplest radical or exponential form as appropriate.
Simplify: \( x^{1/2} \cdot x^{1/2} \)
$$x^{1/2 + 1/2} = x^1 = x$$
Answer: \( x \)
Simplify: \( (y^{1/3})^3 \)
$$y^{(1/3)\cdot 3} = y^1 = y$$
Answer: \( y \)
Simplify: \( a^{3/4} \cdot a^{1/4} \)
$$a^{3/4 + 1/4} = a^1 = a$$
Answer: \( a \)
Simplify: \( (8^{1/3})^2 \)
$$8^{1/3} = 2 \), so \( 2^2 = 4 \)
Or \( 8^{2/3} = (2^3)^{2/3} = 2^2 = 4 \)
Answer: \( 4 \)
Simplify: \( x^{2/3} \cdot x^{4/3} \)
$$x^{2/3 + 4/3} = x^{6/3} = x^2$$
Answer: \( x^2 \)
Simplify: \( 5a^{1/2} \cdot a^{3/4} \)
Add the exponents (common denominator 4):
$$5a^{1/2 + 3/4} = 5a^{2/4 + 3/4} = 5a^{5/4}$$
Answer: \( 5a^{5/4} \)
Simplify: \( 2x^{2/3} \cdot x^{1/2} \)
Add the exponents (common denominator 6):
$$2x^{2/3 + 1/2} = 2x^{4/6 + 3/6} = 2x^{7/6}$$
Answer: \( 2x^{7/6} \)
Simplify: \( 3b^{3/4} \cdot b^{1/2} \)
Add the exponents (common denominator 4):
$$3b^{3/4 + 1/2} = 3b^{3/4 + 2/4} = 3b^{5/4}$$
Answer: \( 3b^{5/4} \)
Simplify: \( (4a^{1/2})(a^{3/2}) \)
$$4 \cdot a^{1/2 + 3/2} = 4a^{4/2} = 4a^2$$
Answer: \( 4a^2 \)
Simplify: \( \dfrac{x^{5/4}}{x^{1/4}} \)
$$x^{5/4 – 1/4} = x^{4/4} = x^1 = x$$
Answer: \( x \)
These problems combine several exponent rules. Simplify completely and write answers with positive exponents only.
Simplify: \( \left(4b^{1/3}\right)\left(b^{2/3}\right) \)
$$4 \cdot b^{1/3 + 2/3} = 4b^{3/3} = 4b$$
Answer: \( 4b \)
Simplify: \( 5d^2 (3d^0)^2 \)
$$d^0 = 1 \), so \( (3 \cdot 1)^2 = 9 \)
$$5d^2 \cdot 9 = 45d^2$$
Answer: \( 45d^2 \)
Simplify with no negative exponents: \( \dfrac{6 \cdot 4^{-2}}{4^{-3}} \)
$$6 \cdot 4^{-2 – (-3)} = 6 \cdot 4^{-2+3} = 6 \cdot 4^1 = 6 \cdot 4 = 24$$
Answer: \( 24 \)
Simplify: \( (2x^3 y)^2 \cdot (3x y^2) \)
$$(2x^3 y)^2 = 4x^6 y^2$$
$$4x^6 y^2 \cdot 3x y^2 = 12x^{7} y^{4}$$
Answer: \( 12x^7 y^4 \)
Simplify: \( \left(9a^{2/3}\right)\left(a^{1/3}\right) \)
$$9 \cdot a^{2/3 + 1/3} = 9a^{3/3} = 9a$$
Answer: \( 9a \)
Need help mastering exponent rules for Algebra 2?