If g(x)=(x^2)/(4)-(1)/(x^2), simplify √(1+[g(x)]^2) |...

Name: Problem 104, Chapter 7: Algebra and Trigonometry
Uploaded: 2021-06-28T19:32:13-08:00
Duration: 6 min 17 s
Channel: Alexis Tomlin
Description: If g(x)=(x^2)/(4)-(1)/(x^2), simplify √(1+[g(x)]^2)

Key Concepts

Absolute Value Consideration in Radical Simplification

When an expression inside a square root is recognized as a perfect square, the result of taking the square root is the absolute value of the underlying expression. This principle reinforces the idea that the square root of a squared term is non-negative, ensuring that cases involving potential negative values are correctly handled.

Manipulating Rational Expressions

This concept focuses on handling expressions that contain variables in the denominator. It involves finding common denominators, combining terms, and simplifying complex algebraic fractions. Such techniques are necessary to prepare expressions for further operations, such as taking square roots or identifying perfect squares.

Squaring and Expanding Functions

This concept involves taking an algebraic expression and raising it to the power of two, which requires expanding the expression using the distributive property or the binomial expansion formula. In the process, each term is squared and cross-products are calculated appropriately, a technique essential for manipulating polynomial and rational expressions.

Recognizing Perfect Square Structures

In algebra, identifying a perfect square is key for simplifying expressions, especially those involving square roots. When an expression under a square root can be rewritten as the square of another expression, it allows for the use of the principle that the square root of a square gives the absolute value of the original expression, thereby greatly simplifying the problem.

Transcript

0:00 Hello.

00:01 So if gfx equals x squared over 4 minus 1 over x squared, simplify the square root of 1 plus g of x squared.

00:13 So all we're going to do is plug in g of x, which is this, into this.

00:23 So we're going to have 1 plus in parentheses x squared over 4 minus 1x squared.

00:34 To the second power or just squared.

00:39 Now we're going to make sure, combine these two together by finding the common denominator, which is 4x squared.

00:51 So to do this, we're gonna multiply x squared to here to the top and bottom, and we're gonna times four on the top and bottom here.

01:02 And we're gonna get x to the fourth, minus 4 and then parentheses square 1 plus all under the square root now we're going to use what we know about exponents and we're going to square the numerator in a denominator so we're going to end up with x to the 4th minus 4 squared and then divide that now 4 x squared to 4x squared to and it's squared, that means that it's going to end up at 16, because 4 times 4 is 16, and then x squared is x to 4th.

01:56 And then don't forget the rest of the problem.

02:03 So now we're going to turn this one into 16x to the 4 over 16x to the 4.

02:11 So we can combine this all together.

02:16 Just copy down what we had before.

02:18 And then when you combine it you will get 16x to the 4th plus x to the 4th minus 4 squared all of that above 16 x to the 4th now the next step is to expand the x to the 4th minus 4 squared which would be x to the 8th minus 8 x to the 4th plus 16 and don't forget the rest so 16 x to the 4th and then all of that is above 16 x to the 4th and then the square root when you simplify this the numerator you'll get 8x to the 4th because 16 x to the 4th minus 8x to the 4th is 8 x to the 4th is 8 x to the 4th and then plus x to the eighth plus 16 over 16 x to the fourth and then the square root...

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