Question

1. When we multiply two powers with the same base, we ____ the exponents. So$3^4\cdot 3^5$ = _______

When we divide two powers with the same base, we ____ the exponents. So$\frac{3^4}{3^2}$ = ______

Short answer

1. When we multiply two powers with the same base, we add the exponents. So$3^4\cdot 3^5=3^9$

When we divide two powers with the same base, we subtract the exponents. So$\frac{3^4}{3^2}=3^2$

Step-by-step solution

(a)Step1 .  Apply the concept of Exponential Notation

If a is any real number and n is a positive integer, then the$n^{th}$power of a is$a^n=\underset{n-times}{\underbrace{a\cdot a\cdot \mathrm{.....}\cdot a}}$

The number a is called thebase, and n is called the exponent.

Step-2 – Apply the concept of Laws of Exponents

• $a^m\cdot a^n=a^{m+n}$
• $\frac{a^m}{a^n}=a^{m-n}$
  

Step-3 – Apply the concept of Laws of Exponents

Since$a^m\cdot a^n=a^{m+n}$ ,when we multiply two powers with the same base, we add the exponents.

So,

$\begin{array}{l}{3}^4\cdot 3^5=3^{4+5}\\ \Rightarrow 3^4\cdot 3^5=3^9\end{array}$

(b)Step1 .  Apply the concept of Exponential Notation

If a is any real number and n is a positive integer, then the$n^{th}$power of a is$a^n=\underset{n-times}{\underbrace{a\cdot a\cdot \mathrm{.....}\cdot a}}$

The number a is called thebase, and n is called the exponent.

Step-2 – Apply the concept of Laws of Exponents

• $a^m\cdot a^n=a^{m+n}$
• $\frac{a^m}{a^n}=a^{m-n}$
  

Step-3 – Apply the concept of Laws of Exponents

Since$\frac{a^m}{a^n}=a^{m-n}$ ,when we divide two powers with the same base, we subtract the exponents.

So,

$\begin{array}{l}\frac{3^4}{3^2}=3^{4-2}\\ \Rightarrow \frac{3^4}{3^2}=3^2\end{array}$