If p > 0 and (p^(m))^(2n)*(p^(2m))^(n)=16^(mn), then what is the value of p ?

Simplify left side of equation: We are given the equation (p^(m))^(2n)*(p^(2m))^(n)=16^(mn) and we need to find the value of p. First, we simplify the left side of the equation using the power of a power rule, which states that (a^b)^c = a^(b*c). (p^(m))^(2n) = p^(m*2n) = p^(2mn) (p^(2m))^(n) = p^(2m*n) = p^(2mn) Now we multiply the two terms together. p^(2mn) * p^(2mn) = p^(2mn + 2mn) = p^(4mn)Multiply the terms together: Next, we simplify the right side of the equation. 16^(mn) can be written as (2^4)^(mn) because 16 is 2 raised to the power of 4. (2^4)^(mn) = 2^(4*mn) = 2^(4mn)Simplify right side of equation: Now we have the equation p^(4mn) = 2^(4mn). Since the exponents of p and 2 are the same, we can equate the bases. p = 2

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

If
p > 0 and
(p^(m))^(2n)*(p^(2m))^(n)=16^(mn), then what is the value of
p ?

If p>0 and $\left(p^{m}\right)^{2 n} \cdot\left(p^{2 m}\right)^{n}=16^{m n}$ , then what is the value of $p$ ?

View step-by-step help

Home

Math Problems

Algebra 1

Compare linear and exponential growth

Full solution

Q. If

p>0

and

\left(p^{m}\right)^{2 n} \cdot\left(p^{2 m}\right)^{n}=16^{m n}

, then what is the value of

p

Simplify left side of equation: We are given the equation $(p^{m})^{2n}\cdot(p^{2m})^{n}=16^{mn}$ and we need to find the value of $p$ . $\newline$ First, we simplify the left side of the equation using the power of a power rule, which states that $(a^{b})^{c} = a^{b\cdot c}$ . $\newline$ $(p^{m})^{2n} = p^{m\cdot 2n} = p^{2mn}$ $\newline$ $(p^{2m})^{n} = p^{2m\cdot n} = p^{2mn}$ $\newline$ Now we multiply the two terms together. $\newline$ $p^{2mn} \cdot p^{2mn} = p^{2mn + 2mn} = p^{4mn}$
Simplify right side of equation: Next, we simplify the right side of the equation. $\newline$ $16^{mn}$ can be written as $(2^4)^{mn}$ because $16$ is $2$ raised to the power of $4$ . $\newline$ $(2^4)^{mn} = 2^{4\cdot mn} = 2^{4mn}$
Solve for $p$ : Equate left and right side of equation to solve for $p$ . $\newline$ Now we have the equation $p^{4mn} = 2^{4mn}$ . $\newline$ Since the exponents of $p$ and $2$ are the same, we can equate the bases. $p = 2$