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Sagot :
To solve the equation [tex]\(32^{2c} = 8^{c+7}\)[/tex], we will follow these steps:
1. Rewrite the numbers as powers of 2:
- [tex]\(32\)[/tex] can be written as [tex]\(2^5\)[/tex].
- [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex].
So, the equation [tex]\(32^{2c} = 8^{c+7}\)[/tex] can be rewritten using these base-2 representations:
[tex]\[ (2^5)^{2c} = (2^3)^{c+7} \][/tex]
2. Simplify the exponents:
Using the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[ 2^{5 \cdot 2c} = 2^{3 \cdot (c+7)} \][/tex]
Simplify the exponents:
[tex]\[ 2^{10c} = 2^{3c + 21} \][/tex]
3. Set the exponents equal to each other:
Since the bases are the same (both are 2), we can set the exponents equal to each other:
[tex]\[ 10c = 3c + 21 \][/tex]
4. Solve for [tex]\(c\)[/tex]:
Isolate [tex]\(c\)[/tex] by subtracting [tex]\(3c\)[/tex] from both sides of the equation:
[tex]\[ 10c - 3c = 21 \][/tex]
[tex]\[ 7c = 21 \][/tex]
Divide both sides by 7:
[tex]\[ c = \frac{21}{7} \][/tex]
[tex]\[ c = 3 \][/tex]
Thus, the value of [tex]\(c\)[/tex] is [tex]\(\boxed{3}\)[/tex].
1. Rewrite the numbers as powers of 2:
- [tex]\(32\)[/tex] can be written as [tex]\(2^5\)[/tex].
- [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex].
So, the equation [tex]\(32^{2c} = 8^{c+7}\)[/tex] can be rewritten using these base-2 representations:
[tex]\[ (2^5)^{2c} = (2^3)^{c+7} \][/tex]
2. Simplify the exponents:
Using the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[ 2^{5 \cdot 2c} = 2^{3 \cdot (c+7)} \][/tex]
Simplify the exponents:
[tex]\[ 2^{10c} = 2^{3c + 21} \][/tex]
3. Set the exponents equal to each other:
Since the bases are the same (both are 2), we can set the exponents equal to each other:
[tex]\[ 10c = 3c + 21 \][/tex]
4. Solve for [tex]\(c\)[/tex]:
Isolate [tex]\(c\)[/tex] by subtracting [tex]\(3c\)[/tex] from both sides of the equation:
[tex]\[ 10c - 3c = 21 \][/tex]
[tex]\[ 7c = 21 \][/tex]
Divide both sides by 7:
[tex]\[ c = \frac{21}{7} \][/tex]
[tex]\[ c = 3 \][/tex]
Thus, the value of [tex]\(c\)[/tex] is [tex]\(\boxed{3}\)[/tex].
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