Sure, let's express each of the given rational numbers in their exponential forms step-by-step.
### (i) [tex]\(\frac{25}{64}\)[/tex]
1. Start by recognizing the numerical value of the fraction:
[tex]\[
\frac{25}{64} = 0.390625
\][/tex]
2. Identify the base numbers that can be rewritten with the same exponent:
[tex]\[
25 = 5^2 \quad \text{and} \quad 64 = 8^2
\][/tex]
3. Rewrite the fraction in terms of its bases:
[tex]\[
\frac{25}{64} = \frac{5^2}{8^2}
\][/tex]
4. Notice that both the numerator and the denominator have the same exponent, so it can be written as:
[tex]\[
\frac{5^2}{8^2} = \left(\frac{5}{8}\right)^2
\][/tex]
Thus, the expression in exponential form is:
[tex]\[
\frac{25}{64} = \left(\frac{5}{8}\right)^2
\][/tex]
### (ii) [tex]\(-\frac{125}{216}\)[/tex]
1. Start by recognizing the numerical value of the fraction:
[tex]\[
-\frac{125}{216} = -0.5787037037037037
\][/tex]
2. Identify the base numbers that can be rewritten with the same exponent:
[tex]\[
125 = 5^3 \quad \text{and} \quad 216 = 6^3
\][/tex]
3. Rewrite the fraction in terms of its bases:
[tex]\[
-\frac{125}{216} = -\frac{5^3}{6^3}
\][/tex]
4. Notice that both the numerator and the denominator have the same exponent, so it can be written as:
[tex]\[
-\frac{5^3}{6^3} = -\left(\frac{5}{6}\right)^3
\][/tex]
Thus, the expression in exponential form is:
[tex]\[
-\frac{125}{216} = -\left(\frac{5}{6}\right)^3
\][/tex]
### (iii) [tex]\(-\frac{343}{729}\)[/tex]
1. Start by recognizing the numerical value of the fraction:
[tex]\[
-\frac{343}{729} = -0.47050754458161864
\][/tex]
2. Identify the base numbers that can be rewritten with the same exponent:
[tex]\[
343 = 7^3 \quad \text{and} \quad 729 = 9^3
\][/tex]
3. Rewrite the fraction in terms of its bases:
[tex]\[
-\frac{343}{729} = -\frac{7^3}{9^3}
\][/tex]
4. Notice that both the numerator and the denominator have the same exponent, so it can be written as:
[tex]\[
-\frac{7^3}{9^3} = -\left(\frac{7}{9}\right)^3
\][/tex]
Thus, the expression in exponential form is:
[tex]\[
-\frac{343}{729} = -\left(\frac{7}{9}\right)^3
\][/tex]
I hope this detailed explanation helps you understand how to express these rational numbers in exponential form!
