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Sagot :
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!
### (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!
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