Question

quiz 1 exponent properties project
e your project on another page and staple it to this page.
mplete about the following exponent properties:

1. product rule: $a^m \cdot a^n = a^{m + n}$

Explanation:

The problem is about explaining the Product Rule of exponents (\(a^m \cdot a^n = a^{m + n}\)). Here's a step - by - step explanation:

Step 1: Understand the meaning of exponents

An expression like \(a^m\) means that the base \(a\) is multiplied by itself \(m\) times. Mathematically, we can write \(a^m=\underbrace{a\times a\times\cdots\times a}_{m\text{ times}}\) and \(a^n=\underbrace{a\times a\times\cdots\times a}_{n\text{ times}}\)

Step 2: Multiply \(a^m\) and \(a^n\)

When we multiply \(a^m\) and \(a^n\), we have \((\underbrace{a\times a\times\cdots\times a}_{m\text{ times}})\times(\underbrace{a\times a\times\cdots\times a}_{n\text{ times}})\)

Step 3: Count the number of multiplications

If we count the total number of times \(a\) is multiplied by itself, we add the number of multiplications from \(a^m\) and \(a^n\). So the total number of multiplications is \(m + n\) times. Which means \(a^m\times a^n=\underbrace{a\times a\times\cdots\times a}_{m + n\text{ times}}=a^{m + n}\)

For example, if \(a = 2\), \(m=3\) and \(n = 2\). Then \(a^m=2^3=2\times2\times2 = 8\), \(a^n=2^2=2\times2 = 4\). \(a^m\times a^n=8\times4 = 32\). And \(a^{m + n}=2^{3+2}=2^5=32\). So \(2^3\times2^2 = 2^{3 + 2}\)

Answer:

The Product Rule of exponents \(a^m\cdot a^n=a^{m + n}\) states that when multiplying two exponential expressions with the same base, we keep the base the same and add the exponents. This is because \(a^m\) represents \(m\) multiplications of \(a\) and \(a^n\) represents \(n\) multiplications of \(a\), and when we multiply them together, we have a total of \(m + n\) multiplications of \(a\)