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The equation $2^c \cdot b^c = 10^c$ is true for all values of $c$ . What is the value of $b$ ?

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Solve equations with variable exponents

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Q. The equation

2^c \cdot b^c = 10^c

is true for all values of

c

. What is the value of

b

?

Recognize Property of Exponents: Recognize that the equation $2^c \times b^c = 10^c$ can be simplified by using the property of exponents that states $(a\times b)^c = a^c \times b^c$ .
Rewrite Equation with 2*5)^c:\ Since \$10^c can be written as $2*5)^c\, we can rewrite the equation as \$2^c * b^c = (2*5)^c$ .
Apply Property of Exponents in Reverse: Apply the property of exponents from Step $1$ in reverse to get $2^c \times b^c = 2^c \times 5^c$ .
Equate Bases with Same Exponent: Since the equation must hold for all values of $c$ , we can equate the bases with the same exponent $c$ . This gives us $b^c = 5^c$ .
Determine Value of $\newline$ $b$ : If $b^c = 5^c$ , then $b$ must be equal to $5$ because if the bases are the same and the exponents are the same, then the numbers themselves must be equal.

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