The equation 2^(c)*b^(c)=10^(c) is true for all values of c. What is the value of b ?

Given Equation Simplification: We are given the equation 2^(c)*b^(c)=10^(c). We need to find the value of b that makes this equation true for all values of c.Expressing 10 as 2*5: First, we can simplify the equation by using the property of exponents that states (a*b)^c = a^c * b^c. We apply this property to the right side of the equation where 10 can be expressed as 2*5. So, we rewrite 10^(c) as (2*5)^(c), which is equal to 2^(c)*5^(c).Equating Bases with Same Exponent: Now we have 2^(c)*b^(c) = 2^(c)*5^(c). 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).Concluding Value of B: Since b^(c) = 5^(c) for all c, we can conclude that b must be equal to 5, because if b were not equal to 5, there would exist some value of c for which b^(c) would not equal 5^(c).

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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

Given Equation Simplification: We are given the equation $2^{c}\cdot b^{c}=10^{c}$ . We need to find the value of $b$ that makes this equation true for all values of $c$ .
Expressing $10$ as $2\times5$ : First, we can simplify the equation by using the property of exponents that states $(a\times b)^c = a^c \times b^c$ . We apply this property to the right side of the equation where $10$ can be expressed as $2\times5$ . So, we rewrite $10^{c}$ as $(2\times5)^{c}$ , which is equal to $2^{c}\times5^{c}$ .
Equating Bases with Same Exponent: Now we have $2^{c}\cdot b^{c} = 2^{c}\cdot 5^{c}$ . Since the equation must hold for all values of $c$ , we can equate the bases with the same exponent $c$ . This gives us $b^{c} = 5^{c}$ .
Concluding Value of B: Since $b^{c} = 5^{c}$ for all $c$ , we can conclude that $b$ must be equal to $5$ , because if $b$ were not equal to $5$ , there would exist some value of $c$ for which $b^{c}$ would not equal $5^{c}$ .