Laws of Exponents

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What is the product rule in exponents?
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The product rule states that when multiplying two exponents with the same base, you add the exponents. For example, a^m * a^n = a^(m+n).
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How do you apply the power rule to exponents?
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The power rule states that when you raise a power to another power, you multiply the exponents. For example, (a^m)^n = a^(m*n).
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What is the quotient rule in exponents?
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The quotient rule states that when dividing exponents with the same base, you subtract the exponents. For example, a^m / a^n = a^(m-n).
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Explain a zero exponent.
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Any number raised to the zero power is equal to 1, except for zero itself. For example, a^0 = 1.
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What happens when you have a negative exponent?
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A negative exponent represents a reciprocal. For example, a^(-n) = 1/a^n.
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How do you simplify (a^m)^0?
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Any nonzero base to the zero power is 1, so (a^m)^0 = 1.
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What is the rule for multiplying two equal bases but with negative exponents?
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Add the exponents, including their negative signs. For example, a^(-m) * a^(-n) = a^(-m-n).
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What does (x^m * y^m) equal?
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(x^m * y^m) is equal to (xy)^m.
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How do you express b^(-n) as a fraction?
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b^(-n) is expressed as 1/b^n.
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What does it mean for exponents to be distributive over multiplication?
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It means that (xy)^n = x^n * y^n.
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If a^5 = 1, what can you say about a?
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a must be equal to 1 if a^5 = 1, assuming a is a positive real number.
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Can you simplify (x^3 / x^3)?
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Yes, (x^3 / x^3) equals x^(3-3) = x^0 = 1.
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What is (k^4)^-1 in simplest form?
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(k^4)^-1 is equal to 1/(k^4).
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How would you simplify ((a^2)^3)^4?
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You multiply the exponents: 2 * 3 * 4 = 24, so ((a^2)^3)^4 = a^24.
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If b^(m/n) = x, how can this be rewritten using roots?
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b^(m/n) can be rewritten as the n-th root of b^m, or (root(n)(b))^m.
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