radicals

[tommy81874]

how do you solve this radical square root 5 of -243x35n15

 

[Denis]

how do you solve this radical square root 5 of -243x35n15

Tommy, ask someone to show you how to post PROPERLY.

What is the "x" and the "n"?
What does "of" mean?
Where is the = sign?

 

[mmm4444bot]

We dont' say "solve" the radical; we say "simplify" the radical.

We don't say "square root 5" of something; we say "5th root" of something.

There are different ways to type a fifth root:

fifthroot(-243 x^35 n^15)

root[5](-243 x^35 n^15)

(-243 x^35 n^15)^(1/5)

[sup:3t9a4flz]5[/sup:3t9a4flz]?(-243 x^35 n^15)

The key is knowing that five identical factors inside the radical sign simplify to one factor outside.

I mean, the concept is the same as for simplify square roots, where pairs of factors underneath the radical sign simplify to one factor outside the radical sign.

EG

fifthroot(x^5 * n^2)

fifthroot(x*x*x*x*x * n*n)

x * fifthroot(n^2)

Five factors inside become one factor outside.

Two factors inside are stuck inside.

-243 is a power of -3.

-243 = (-3)^5

(-3)^5 is five identical factors of -3, yes ?

I mean, (-3)^5 = (-3)(-3)(-3)(-3)(-3)

The fifth root of (-3)(-3)(-3)(-3)(-3) simplifies to (-3).

In other words, [sup:3t9a4flz]5[/sup:3t9a4flz]?(-243) = -3.

x^35 is five identical factors of x^7.

(x^7)(x^7)(x^7)(x^7)(x^7) = x^(7 + 7 + 7 + 7 + 7) = x^35

Those five factors inside the radical simplify to one factor outside, so the fifth root of x^35 simplifies to x^7.

n^15 is (n^3)(n^3)(n^3)(n^3)(n^3), so the fifth root of n^15 simplifies to n^3.

The given expression simplifies to:

-3 x^7 n^3

After you learn to express nth roots exponentially, using the exponent 1/n, these exercises become easier to simplify.

Written exponentially, the fifth root is: ([-3]^5 x^35 n^15)^(1/5)

We simply divide each exponent 5, 35, and 15 by 5 because there's a property for that:

[-3]^(5/5) x^(35/5) n^(15/5)

-3 x^7 n^3

Different approach; same result.

There are many different approaches. I posted what first comes to my mind.

Make sure that you understand the meaning of roots, and use properties of radicals to assist your understanding.

Cheers ~ Mark

 

[BigGlenntheHeavy]

\(\displaystyle Another \ way.\)

\(\displaystyle \sqrt[5]{-243x^{35}n^{15}} \ = \ \sqrt[5]{(-3)^5x^{35}n^{15}} \ = \ [(-3)^5x^{35}n^{15}]^{1/5} \ = \ [(-3)^5]^{1/5}[x^{35}]^{1/5}[n^{15}]^{1/5}\)

\(\displaystyle = \ (-3)x^7n^3 \ = \ -3x^7n^3.\)