Question

example 3.1 - i: evaluating exponential expressions using a calculator
evaluate the following using a calculator:
a. $5^{6}$
b. $(\frac{3}{2})^{3}$
c. $(1.02)^{4}$
for problems 1 and 2, identify the values in the empty cells for repeated multiplication, base, exponent, and power (exponential notation).
question 1:

question	repeated multiplication	base	exponent	power (exponential notation)
b.				$9^{5}$
c.		$3$	$4$
d.	$\frac{2}{5}\times\frac{2}{5}\times\frac{2}{5}\times\frac{2}{5}\times\frac{2}{5}\times\frac{2}{5}$
e.				$(\frac{5}{7})^{5}$
f.		$(\frac{4}{7})$	$3$
g.	$(1.15)\times(1.15)\times(1.15)\times(1.15)$
h.				$(1.6)^{3}$
i.		$(1.25)$	$5$

Explanation:

Step1: Recall exponential - notation rules

In \(a^n\), \(a\) is the base and \(n\) is the exponent. Repeated - multiplication form of \(a^n\) is \(a\times a\times\cdots\times a\) (\(n\) times).

Step2: Evaluate the exponential expressions in the first part

a. For \(5^6\)

Using a calculator, \(5^6=5\times5\times5\times5\times5\times5 = 15625\).

b. For \((\frac{3}{2})^3\)

\((\frac{3}{2})^3=\frac{3}{2}\times\frac{3}{2}\times\frac{3}{2}=\frac{27}{8}=3.375\).

c. For \((1.02)^4\)

\((1.02)^4 = 1.02\times1.02\times1.02\times1.02=1.08243216\).

Step3: Fill in the table for repeated - multiplication, base, exponent, and power

a.

Base: \(7\), Exponent: \(4\), Power: \(7^4\)

b.

Repeated Multiplication: \(9\times9\times9\times9\times9\), Base: \(9\), Exponent: \(5\)

c.

Repeated Multiplication: \(3\times3\times3\times3\), Power: \(3^4\)

d.

Base: \(\frac{2}{5}\), Exponent: \(6\), Power: \((\frac{2}{5})^6\)

e.

Repeated Multiplication: \(\frac{5}{7}\times\frac{5}{7}\times\frac{5}{7}\times\frac{5}{7}\times\frac{5}{7}\), Base: \(\frac{5}{7}\), Exponent: \(5\)

f.

Repeated Multiplication: \(\frac{4}{7}\times\frac{4}{7}\times\frac{4}{7}\), Power: \((\frac{4}{7})^3\)

g.

Base: \(1.15\), Exponent: \(4\), Power: \((1.15)^4\)

h.

Repeated Multiplication: \(1.6\times1.6\times1.6\), Base: \(1.6\), Exponent: \(3\)

i.

Repeated Multiplication: \(1.25\times1.25\times1.25\times1.25\times1.25\), Power: \((1.25)^5\)

Answer:

a. \(15625\)
b. \(3.375\)
c. \(1.08243216\)
Table - filled:

Question	Repeated Multiplication	Base	Exponent	Power (Exponential Notation)
b.	\(9\times9\times9\times9\times9\)	\(9\)	\(5\)	\(9^5\)
c.	\(3\times3\times3\times3\)	\(3\)	\(4\)	\(3^4\)
d.	\(\frac{2}{5}\times\frac{2}{5}\times\frac{2}{5}\times\frac{2}{5}\times\frac{2}{5}\times\frac{2}{5}\)	\(\frac{2}{5}\)	\(6\)	\((\frac{2}{5})^6\)
e.	\(\frac{5}{7}\times\frac{5}{7}\times\frac{5}{7}\times\frac{5}{7}\times\frac{5}{7}\)	\(\frac{5}{7}\)	\(5\)	\((\frac{5}{7})^5\)
f.	\(\frac{4}{7}\times\frac{4}{7}\times\frac{4}{7}\)	\(\frac{4}{7}\)	\(3\)	\((\frac{4}{7})^3\)
g.	\((1.15)\times(1.15)\times(1.15)\times(1.15)\)	\(1.15\)	\(4\)	\((1.15)^4\)
h.	\(1.6\times1.6\times1.6\)	\(1.6\)	\(3\)	\((1.6)^3\)
i.	\(1.25\times1.25\times1.25\times1.25\times1.25\)	\(1.25\)	\(5\)	\((1.25)^5\)