Question

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

Short answer

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: \[(5^2)^3 = 5^{2\cdot3} = 5^6\]

Key concepts

Understanding the Product Rule

The product rule is one of the foundational laws of indices that helps us handle exponential expressions with ease. When you multiply two expressions that have the same base but different powers, you simply add the exponents together. This rule can be expressed mathematically as: \[ a^m \cdot a^n = a^{m+n} \]Here's what happens with this rule:

• Base remains the same: The base "a" does not change during multiplication.
• Exponents add up: Only the exponents are added together.

For instance, let's consider \( 2^3 \cdot 2^4 \).

• The base here is \(2\), so it remains unchanged.
• By adding the exponents (3 + 4), we find that \(2^3 \cdot 2^4 = 2^7\).

This simple operation reduces what could be a complex multiplication issue into a straightforward addition problem. The product rule makes calculations much quicker and straightforward when dealing with powers.

Explaining the Quotient Rule

The quotient rule is another crucial law of indices. It helps us simplify expressions where exponential terms with the same base are divided by each other. The quotient rule tells us to subtract the exponent of the divisor from the exponent of the dividend. Mathematically, it is written as: \[ \frac{a^m}{a^n} = a^{m-n} \] Here's how this rule works:

• Consistency of base: Both the numerator and the denominator must have the same base.
• Subtract exponents: Subtract the lower (or divisor's) exponent from the higher (or dividend's) exponent.

Consider the example \( \frac{3^5}{3^2} \):

• Keep the base \(3\) the same.
• Subtract the exponents (5 - 2) to get \(3^3\).

This rule simplifies division with exponents and helps avoid long division calculations. Using the quotient rule keeps the process clean and efficient.

Decoding the Power Rule

The power rule is a handy tool when dealing with powers raised to another power. The idea is that we multiply the exponents together whenever exponents are involved in such a layered manner. This law is expressed as: \[ (a^m)^n = a^{m \cdot n} \] How the power rule operates:

• Same base principle: The base "a" remains unchanged, as with the other rules.
• Multiply exponents: The exponents, "m" and "n," are multiplied with each other.

Let's see it in action with \( (5^2)^3 \):

• You maintain the base of \(5\), just as before.
• By multiplying the exponents (2 \times 3), the expression simplifies to \(5^6\).

The power rule makes it incredibly easy to tackle more complex equations with power-on-power scenarios. It eliminates the need to expand or write out massive calculations.