Chapter 1.2, Problem 1E

1. (a) Using exponential notation, we can write the product 5 · 5 · 5 · 5 · 5 · 5 as __________.
2. (b) In the expression 3⁴ the number 3 is called the __________, and the number 4 is called the __________.

To determine

To fill: The blank in the sentence “Using exponential notation, we can write the product $5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5$ as ________”.

Answer to Problem 1E

The complete sentence is “Using exponential notation, we can write the product $5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5$ as $\underset{\_}{5^6}$”.

Explanation of Solution

Definition used:

Exponential notation:

“If a is any real number and n is a positive integer, then the $n^{th}$ power of a is $a^n=\underset{n\text{ factors}}{\underbrace{a\cdot a\cdot a\cdots a}}$ where the number a is called as the base and n is called the exponent”.

Calculation:

Consider the given product which is $5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5$.

Observe that, in the given product the number 5 is multiplied 6 times.

Therefore, by the above definition it can be noted that, here the base is 5 and the exponent is 6.

Therefore, substitute $a=5\text{ and }n=6$ in $a^n$ and obtain the exponential notation as shown below.

$\begin{array}{l}a\cdot a\cdot a\cdot a\cdot a\cdot a\dots =a^n\\ 5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5=5^6\end{array}$

Thus, the exponential notation of the product $5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5$ is $\underset{\_}{5^6}$.

Therefore, the complete sentence is “Using exponential notation, we can write the product $5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5$ as $\underset{\_}{5^6}$”.

To determine

To fill: The blank in the sentence “In the expression $3^4$ the number 3 is called ____ and the number 4 is called _____”.

Answer to Problem 1E

The complete sentence is “In the expression $3^4$ the number 3 is called $\underset{\_}{\text{Base}}$ and the number 4 is called $\underset{\_}{\text{Exponent}}$”.

Explanation of Solution

The given expression is $3^4$.

Use the definition given in part (a) and rewrite $3^4$ as shown below.

$3^4=3\cdot 3\cdot 3\cdot 3$

That is, here the number 3 is repeated 4 times.

Therefore, by the definition of exponential notation it can be noted that the number 3 is the base and the number 4 is the exponent.

Thus, the complete statement is “In the expression $3^4$ the number 3 is called $\underset{\_}{\text{Base}}$ and the number 4 is called $\underset{\_}{\text{Exponent}}$”.