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When you multiply powers that have the same base, you add the exponents. This is known as the "product rule of exponents." The logic behind this rule is simple: when you multiply like bases, you're essentially multiplying the same number by itself multiple times. Thus, you just count up all the times it's multiplied.
For example, consider the expression \( a^4 \cdot a^6 \). This means you are multiplying "a" four times in the first power and six more times in the second power. Instead of writing it out as \( a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \), you can simply add the exponents:

• Take the exponents 4 and 6.
• Add them together: \( 4 + 6 = 10 \).

Thus, \( a^4 \cdot a^6 = a^{10} \). This makes life much easier when working with large exponents!

The quotient rule of exponents states that when you divide powers with the same base, you need to subtract the exponents. This rule comes into play when simplifying expressions involving division of like bases.
Consider the case where you have \( a^{10} \div a^{2} \). Here's the approach you take:

• Identify the exponents (10 and 2 in this case) of the terms being divided.
• Subtract the exponent in the divisor from the exponent in the dividend: \( 10 - 2 \).

This calculation gives you \( a^8 \). So, dividing \( a^{10} \) by \( a^2 \) simplifies to \( a^{8} \). This rule helps simplify expressions without having to expand them fully.

Simplifying expressions involving exponents requires the application of both the product and quotient rules effectively. It involves reducing complex expressions to their simplest form.
Take the expression \( a^4 \cdot a^6 \div a^2 \). Here's a step-by-step breakdown:

• First, apply the product rule to \( a^4 \cdot a^6 \). Add the exponents: \( 4 + 6 = 10 \), resulting in \( a^{10} \).
• Next, apply the quotient rule to \( a^{10} \div a^{2} \). Subtract the exponents: \( 10 - 2 = 8 \), leading to the simplified expression \( a^{8} \).

This process of using both rules ensures that expressions are reduced to their simplest and most efficient forms, saving time and effort in complex calculations.