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Questions and Answers
Simplify the expression: $y^4 \cdot y^{-2} \cdot y^3$
Simplify the expression: $y^4 \cdot y^{-2} \cdot y^3$
- $y^9$
- $y^5$ (correct)
- $y^{-24}$
- $y^{-1}$
According to the quotient of powers rule, $\frac{5^3}{5^3} = 0$
According to the quotient of powers rule, $\frac{5^3}{5^3} = 0$
False (B)
Which expression is equivalent to $(z^{-2})^4$?
Which expression is equivalent to $(z^{-2})^4$?
- $\frac{1}{z^8}$ (correct)
- $z^2$
- $z^8$
- $\frac{1}{z^2}$
According to the zero exponent rule, any non-zero number raised to the power of zero is equal to ______.
According to the zero exponent rule, any non-zero number raised to the power of zero is equal to ______.
Using the negative exponent rule, express $4^{-3}$ as a fraction.
Using the negative exponent rule, express $4^{-3}$ as a fraction.
Simplify the expression $\frac{a^7 \cdot b^3}{a^2 \cdot b}$.
Simplify the expression $\frac{a^7 \cdot b^3}{a^2 \cdot b}$.
The product of powers rule can only be applied if the exponents are the same.
The product of powers rule can only be applied if the exponents are the same.
Which of the following is equivalent to $(5x)^0 + (5x^0)$?
Which of the following is equivalent to $(5x)^0 + (5x^0)$?
$\frac{x^4}{x^{-2}}$ is equivalent to $x$ to the power of ______.
$\frac{x^4}{x^{-2}}$ is equivalent to $x$ to the power of ______.
Match each expression with its simplified form:
Match each expression with its simplified form:
Flashcards
Product of Powers Rule
Product of Powers Rule
When multiplying powers with the same base, add the exponents: a^m * a^n = a^(m+n)
Quotient of Powers Rule
Quotient of Powers Rule
When dividing powers with the same base, subtract the exponents: a^m / a^n = a^(m-n)
Power of a Power Rule
Power of a Power Rule
When raising a power to another power, multiply the exponents: (a^m)^n = a^(m*n)
Zero Exponent Rule
Zero Exponent Rule
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Negative Exponent Rule
Negative Exponent Rule
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Index Laws Definition
Index Laws Definition
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Study Notes
- Index laws are a set of rules that allow simplification of expressions involving exponents
- They provide shortcuts for multiplication, division, and exponentiation of powers with the same base
Product of Powers
- When multiplying powers with the same base, add the exponents: (a^m \cdot a^n = a^{m+n})
- Example: (x^2 \cdot x^3 = x^{2+3} = x^5)
- This law arises from the definition of exponents as repeated multiplication
- E.g., (x^2 \cdot x^3 = (x \cdot x) \cdot (x \cdot x \cdot x) = x \cdot x \cdot x \cdot x \cdot x = x^5)
- The base, (a), must be the same for this rule to apply
- This rule can be extended to multiple terms: (a^m \cdot a^n \cdot a^p = a^{m+n+p})
Quotient of Powers
- When dividing powers with the same base, subtract the exponents: (\frac{a^m}{a^n} = a^{m-n})
- Example: (\frac{x^5}{x^2} = x^{5-2} = x^3)
- This law is the inverse of the product of powers rule
- E.g., (\frac{x^5}{x^2} = \frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x} = x \cdot x \cdot x = x^3)
- The base, (a), must be the same for this rule to apply
- Care must be taken to subtract the exponents in the correct order (numerator exponent minus denominator exponent)
Power of a Power
- When raising a power to another power, multiply the exponents: ((a^m)^n = a^{m \cdot n})
- Example: ((x^2)^3 = x^{2 \cdot 3} = x^6)
- This law can be visualized as repeated exponentiation
- E.g., ((x^2)^3 = x^2 \cdot x^2 \cdot x^2 = x^{2+2+2} = x^6)
- This rule can also be applied when there are multiple exponents: (((a^m)^n)^p = a^{m \cdot n \cdot p})
Zero Exponent Rule
- Any non-zero number raised to the power of zero is equal to 1: (a^0 = 1) (where (a \neq 0))
- Example: (5^0 = 1), (x^0 = 1) (if (x \neq 0))
- This rule can be derived from the quotient of powers rule
- E.g., (\frac{a^m}{a^m} = a^{m-m} = a^0), but (\frac{a^m}{a^m}) also equals 1, therefore (a^0 = 1)
- The case of (0^0) is generally undefined
Negative Exponent Rule
- A number raised to a negative exponent is equal to the reciprocal of that number raised to the positive exponent: (a^{-n} = \frac{1}{a^n})
- Example: (x^{-3} = \frac{1}{x^3}), (2^{-2} = \frac{1}{2^2} = \frac{1}{4})
- This rule can be derived from the quotient of powers rule
- E.g., (\frac{a^0}{a^n} = a^{0-n} = a^{-n}), but (\frac{a^0}{a^n} = \frac{1}{a^n}), therefore (a^{-n} = \frac{1}{a^n})
- It provides a way to express reciprocals using exponents
- Note that (a) cannot be zero
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