Adding Indices

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More Lessons for Arithmetic
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When multiplying numbers in exponent notation with the same base, we can add the exponents.

Consider:

a² × a³ = (a × a) × (a × a × a)
= a^{2 + 3}
= a⁵

This is the first law of exponents:
a^m × aⁿ = a^{m + n}

Example:

Simplify the following; give your answers in exponent form

a) 3³ × 3²
b) x⁵ × x³

Solution:

a) 3³ × 3² = 3^{3 + 2}
= 3⁵
b) x⁵ × x³ = x^{5 + 3}
= x⁸

Common Error : The first law of exponents does NOT apply to addition of numbers in exponent notation.

a^m + aⁿ ≠ a^{m + n}

Scroll down the page for more examples and solutions of the first law of exponents and also the other laws of exponents.

Index rules - add and subtract indices
Basic look at the first two index laws

• Show Step-by-step Solutions

Properties of Exponents
The Product Rule: a^m × aⁿ = a^{m + n}
The Quotient Rule: a^m ÷ aⁿ = a^{m - n}
The Power Rule: (a^m)ⁿ = a^mn
Example:
Simplify
(3ab⁴)(2a²b)³
(x²y⁴/2xz)³

• Show Step-by-step Solutions

Example of adding exponents
(2a⁴)(5¹¹)

• Show Step-by-step Solutions

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