Solve each equation. See Example 7.(x-3)^2/5=(4x)^1/5

Question

Accepted Answer

Hey, everyone here we are asked to solve for X in the following equation where we have the quantity of X minus six raised to the power of 2/5 is equal to the quantity of eight X raised to the power of 1/5. So before we begin solving for X, we need to recall the power to a power rule for exponents which tells us that A to the power of N raised to the power of M, is it equivalent to A to the power of end times? M? Now looking back at our given equation, we want to eliminate the radicals in both of our terms, which means we want to get rid of the fractions in the exponents. So we see that both exponents have a denominator of five. So we want to raise both sides to the power of five to get rid of the denominator. So now simplifying, we have the quantity of X minus six raised to the power of two is equal to the quantity of eight X raised to the power of one. So we see that the quantity of A X raised to the power of one is the same as eight X. So we can just keep it as that. And now we want to get all of our terms our X terms on the same side of our equation. So, so we just want to subtract eight X from both sides. And so now we have the quantity of X minus six squared minus eight X is equal to zero. Well, now we want to factor out the, the quantity of X minus six squared. So we can start simplifying our X terms. So factoring it out, we have X squared minus 12 X plus six minus eight, X is equal to zero. Now, combining like terms, we have X squared minus 20 X plus 36 is equal to zero. Now, we want to factor this rewritten form of our equation so that we can find our values for X. So factoring we have the quantity of X minus two times the quantity of X minus 18 is equal to zero. And now we can apply the zero factor property which tells us that we can take each part of the expression, each factor and set it individually equal to zero. So we can start with our first part of the expression X minus two and again, set it equal to zero. Now we can just solve for X by adding two To, to both sides. And we see that one of our solutions for X is just positive two. And now we can repeat this with the other part of our expression which is X -18. And again, we can set this equal to zero. Now, solving for X by adding 18 to both sides, we see that the other solution for X is 18. So now we have found two values for X or two solutions for X which are two and 18. And with these two values, we are left with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation. See Example 7.(x-3)^2/5=(4x)^1/5

Key Concepts

Exponents and Roots

Quadratic Equations

Isolating Variables

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