What is the expression for f(x) when we rewrite (64^(x))/(4^(5x-1)) as 4^(f(x)) ? f(x)=

Express as Power of 4: First, we need to express 64 as a power of 4 because the denominator is already in base 4. We know that 64 is 4 cubed, or 4^3. So we can rewrite 64^x as (4^3)^x.Apply Power of a Power Rule: Now we apply the power of a power rule, which states that (a^b)^c = a^(b*c). So (4^3)^x becomes 4^(3x).Rewrite with New Base Exponent: Next, we rewrite the original expression with the new base 4 exponent in the numerator: (4^(3x))/(4^(5x-1)).Use Quotient of Powers Rule: Now we use the quotient of powers rule, which states that a^(m)/a^(n) = a^(m-n). So we subtract the exponent in the denominator from the exponent in the numerator: 4^(3x) - 4^(5x-1) becomes 4^(3x - (5x-1)).Simplify Exponent: We simplify the exponent by distributing the negative sign inside the parentheses: 3x - (5x-1) becomes 3x - 5x + 1.Combine Like Terms: Now we combine like terms in the exponent: 3x - 5x + 1 becomes -2x + 1.Final Expression: Finally, we have the expression for f(x) as the exponent of 4 that we've simplified: f(x) = -2x + 1.

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What is the expression for
f(x) when we rewrite
(64^(x))/(4^(5x-1)) as
4^(f(x)) ?

f(x)=

What is the expression for $f(x)$ when we rewrite $\frac{64^{x}}{4^{5 x-1}}$ as $4^{f(x)}$ ? $\newline$ $f(x)=$

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Algebra 1

Compare linear and exponential growth

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Q. What is the expression for

f(x)

when we rewrite

\frac{64^{x}}{4^{5 x-1}}

4^{f(x)}

\newline

f(x)=

Express as Power of $4$ : First, we need to express $64$ as a power of $4$ because the denominator is already in base $4$ . We know that $64$ is $4$ cubed, or $4^3$ . So we can rewrite $64^x$ as $(4^3)^x$ .
Apply Power of a Power Rule: Now we apply the power of a power rule, which states that a^b)^c = a^{b*c}\. So $\(4^ $3$ )^x\ becomes (4$^{ $3$ x}\.
Rewrite with New Base Exponent: Next, we rewrite the original expression with the new base $4$ exponent in the numerator: $(4^{3x})/(4^{5x-1})$ .
Use Quotient of Powers Rule: Now we use the quotient of powers rule, which states that $a^{m}/a^{n} = a^{m-n}$ . So we subtract the exponent in the denominator from the exponent in the numerator: $4^{3x} - 4^{5x-1}$ becomes $4^{3x - (5x-1)}$ .
Simplify Exponent: We simplify the exponent by distributing the negative sign inside the parentheses: $3x - (5x-1)$ becomes $3x - 5x + 1$ .
Combine Like Terms: Now we combine like terms in the exponent: $3x - 5x + 1$ becomes $-2x + 1$ .
Final Expression: Finally, we have the expression for $f(x)$ as the exponent of $4$ that we've simplified: $f(x) = -2x + 1$ .