Chapter 5: Problem 1

Write down the three laws of indices and give a numerical example illustrating each.

Short Answer

Expert verified

Question: State and provide a numerical example for each of the three laws of indices. Answer: The three laws of indices are: 1. Product Rule: When multiplying two powers with the same base, we add the exponents. Example: \(2^3 \cdot 2^4 = 2^{3+4} = 2^7\) 2. Quotient Rule: When dividing two powers with the same base, we subtract the exponent of the divisor from the exponent of the dividend. Example: \(\frac{3^5}{3^2} = 3^{5-2} = 3^3\) 3. Power Rule: When raising a power to another power, we multiply the exponents. Example: \((5^2)^3 = 5^{2\cdot3} = 5^6\)

Step by step solution

Law 1: Product Rule

The product rule states that when multiplying two powers with the same base, we add the exponents. The rule is: \[a^m \cdot a^n = a^{m+n}\] #tag_example# Let's take a numerical example: \[2^3 \cdot 2^4 = 2^{3+4} = 2^7\]

Law 2: Quotient Rule

The quotient rule states that when dividing two powers with the same base, we subtract the exponent of the divisor from the exponent of the dividend. The rule is: \[\frac{a^m}{a^n} = a^{m-n}\] #tag_example# Let's take a numerical example: \[\frac{3^5}{3^2} = 3^{5-2} = 3^3\]

Law 3: Power Rule

The power rule states that when raising a power to another power, we multiply the exponents. The rule is: \[(a^m)^n = a^{mn}\] #tag_example# Let's take a numerical example: \[(5^2)^3 = 5^{2\cdot3} = 5^6\]

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Write down the three laws of indices and give a numerical example illustrating each.

Short Answer

Step by step solution

Law 1: Product Rule

Law 2: Quotient Rule

Law 3: Power Rule

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