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Questions and Answers
What is the result of multiplying the expressions $3^5$ and $3^2$ using the Product of Powers rule?
$3^7$
If you raise the expression $x^4$ to the power of 3, what is the simplified form using the Power of a Power rule?
$x^{12}$
Using the Power of a Product rule, simplify the expression $(4 imes 5)^2$.
$20^2$ or $100$
Explain the Zero Exponent Rule using $7^0$ as an example.
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What would be the result of the expression $2^3 imes 2^{-3}$ applying the Product of Powers rule?
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Study Notes
Laws of Indices
1. Product of Powers
- When multiplying two powers with the same base, add the exponents:
- Formula: ( a^m \times a^n = a^{m+n} )
- Example: ( 2^3 \times 2^4 = 2^{3+4} = 2^7 )
2. Power of a Power
- When raising a power to another power, multiply the exponents:
- Formula: ( (a^m)^n = a^{m \cdot n} )
- Example: ( (3^2)^4 = 3^{2 \cdot 4} = 3^8 )
3. Power of a Product
- When raising a product to a power, apply the exponent to each factor in the product:
- Formula: ( (ab)^n = a^n \times b^n )
- Example: ( (2 \times 3)^3 = 2^3 \times 3^3 = 8 \times 27 = 216 )
4. Zero Exponent Rule
- Any non-zero base raised to the power of zero is equal to one:
- Formula: ( a^0 = 1 ) (for ( a \neq 0 ))
- Example: ( 5^0 = 1 )
Laws of Indices
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Product of Powers: When multiplying powers with the same base, the exponents are added together. For example, multiplying ( 2^3 ) by ( 2^4 ) combines the exponents resulting in ( 2^{3+4} = 2^7 ).
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Power of a Power: To raise a power to another power, multiply the exponents. For instance, ( (3^2)^4 ) simplifies to ( 3^{2 \cdot 4} = 3^8 ).
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Power of a Product: When raising a product to a power, the exponent is distributed to each factor. For example, ( (2 \times 3)^3 ) becomes ( 2^3 \times 3^3 ), ultimately calculating to ( 8 \times 27 = 216 ).
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Zero Exponent Rule: Any non-zero base raised to the exponent of zero equals one. For example, ( 5^0 = 1 ) confirms this rule, emphasizing that only non-zero bases apply.
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Description
Test your understanding of the Laws of Indices, including the Product of Powers, Power of a Power, and Power of a Product. This quiz will help reinforce your knowledge with examples and applications of these essential mathematical concepts.
