Let's solve the questions one by one and understand the detailed solution:
### Question 1:
Determine the value of [tex]\(\frac{(-7)^9}{(-7)^6}\)[/tex]:
Using the property of exponents:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
For [tex]\(\frac{(-7)^9}{(-7)^6}\)[/tex], we have:
[tex]\[
(-7)^{9-6} = (-7)^3
\][/tex]
Now, calculate [tex]\((-7)^3\)[/tex]:
[tex]\[
(-7) \times (-7) \times (-7) = -343
\][/tex]
Thus, the correct option is:
b. [tex]\(-343\)[/tex]
### Question 2:
Evaluate [tex]\(\left(a^x \times a^y\right)^z\)[/tex]:
Using the properties of exponents, we know that:
[tex]\[
\left(a^m \times a^n\right)^k = a^{(m+n)k}
\][/tex]
For [tex]\(\left(a^x \times a^y\right)^z\)[/tex], we can rewrite it as:
[tex]\[
a^{(x+y)z}
\][/tex]
Thus, the correct option is:
d. [tex]\(a^{(x+y)z}\)[/tex]
### Question 3:
Determine the value of [tex]\(4^8 \div 4^{11}\)[/tex]:
Using the property of exponents:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
For [tex]\(4^8 \div 4^{11}\)[/tex], we have:
[tex]\[
4^{8-11} = 4^{-3}
\][/tex]
Converting [tex]\(4^{-3}\)[/tex] to a fraction:
[tex]\[
4^{-3} = \frac{1}{4^3}
\][/tex]
Calculate [tex]\(4^3\)[/tex]:
[tex]\[
4^3 = 64
\][/tex]
Thus, [tex]\(4^{-3} = \frac{1}{64}\)[/tex], and the correct option is:
c. [tex]\(\frac{1}{64}\)[/tex]
### Question 4:
Express 0.0000000531 in standard form:
To convert a number to its standard form, we need to write it as a product of a number between 1 and 10 and a power of 10.
The number 0.0000000531 can be written as:
[tex]\[
5.31 \times 10^{-8}
\][/tex]
Thus, the correct option is:
a. [tex]\(5.31 \times 10^{-8}\)[/tex]
So, the final answers are:
1. b. [tex]\(-343\)[/tex]
2. d. [tex]\(a^{(x+y)z}\)[/tex]
3. c. [tex]\(\frac{1}{64}\)[/tex]
4. a. [tex]\(5.31 \times 10^{-8}\)[/tex]
