How do you rationalize the denominator and simplify #1/(1-8sqrt2)#?

Question

Isaiah Anthony · Accepted Answer

I believe this should be simplified as #(-(8sqrt2+1))/127#.

To rationalize the denominator, you must multiply the term that has the #sqrt# by itself, to move it to the numerator. So: #=>##1/(1-8*sqrt2)*8sqrt2# This will give: #=>##(8sqrt2+1)/(1-(8sqrt2)^2# #(8sqrt2)^2=64*2=128# #=>##(8sqrt2+1)/(1-128)# #=>##(8sqrt2+1)/-127# The negative cam also be moved to the top, for: #=>##(-(8sqrt2+1))/127#

Nathan Axel · Answer

#(-1-8sqrt2)/127#

Multiply the numerator and the denominator by the surd (to undo the surd) and the negative of the extra value.

#1/(1-8sqrt2# x #(-1+8sqrt2)/(-1+8sqrt2#

#(1(1+8sqrt2))/((1-8sqrt2)(1+8sqrt2)#

Expand brackets. Use the FOIL rule for the denominator.

#(1+8sqrt2)/-127#

You could simplify further by taking the negative of the denominator and apply it to the numerator.

#(-1-8sqrt2)/127#

Genesis Allen · Answer

undefined To rationalize the denominator and simplify the expression 1/(1-8sqrt2), we can multiply both the numerator and denominator by the conjugate of the denominator, which is (1+8sqrt2). This will eliminate the square root in the denominator.

By multiplying the numerator and denominator by (1+8sqrt2), we get:

1/(1-8sqrt2) * (1+8sqrt2)/(1+8sqrt2)

Simplifying this expression, we have:

(1 * (1+8sqrt2)) / ((1-8sqrt2) * (1+8sqrt2))

Expanding the numerator and denominator, we get:

(1 + 8sqrt2) / (1 - 64 * 2)

Simplifying further, we have:

(1 + 8sqrt2) / (1 - 128)

Finally, simplifying the denominator, we get:

(1 + 8sqrt2) / (-127)

Therefore, the rationalized and simplified form of 1/(1-8sqrt2) is (1 + 8sqrt2) / (-127).