Solve the equations by introducing a substitution that transforms these equations to quadratic f

Solve the equations by introducing a substitution that...

Key Concepts

Substitution Method

This technique involves replacing a complicated expression with a simpler variable to simplify the equation. By making a suitable substitution, the original equation is transformed into a familiar form, such as a quadratic equation, which is easier to solve.

Exponent Laws

The properties of exponents allow us to rewrite and manipulate expressions involving exponents. For example, rewriting an exponent as a square of another term can facilitate the use of substitution by matching the power structure of a quadratic expression.

Quadratic Equation Transformation

A quadratic equation is a polynomial equation of degree two, which is often easier to solve using standard methods like factoring, completing the square, or the quadratic formula. Through substitution and the application of exponent laws, a more complex equation can be recast into this quadratic form, making the solution process more straightforward.

Transcript

00:01 In this activity, we're being asked to solve an expression, x plus 2 to the 4 thirds power, equals 16, which means find the value of x that makes the statement true.

00:22 The confusing part is we've got this 4 thirds exponent.

00:26 The instructions ask us to rewrite this so that it's in a quadratic form.

00:30 Well, using the properties of exponents, i can take x plus 2 to the 4 thirds power.

00:39 I can use the power property of exponents to rewrite this as x plus 2 to the 2 thirds power, all squared.

00:58 So instead of 4 thirds power, i've got 2 times 2 thirds power broken up into 2 thirds to the second power.

01:09 So that makes this a quadratic with the two pieces that are being multiplied together, being raised to the two -thirds power.

01:23 But that still allows me to get rid of the squared by taking the square root of both sides.

01:31 So i'm going to take the square root of each side.

01:38 That leaves me on one side with x plus 2 to the two -thirds power.

01:51 And on the other side, the square root of 16 can be positive 4 or negative 4.

02:00 But if i look here, this is coming from something that's being squared itself.

02:05 If i square something, i am not going to get a negative 4.

02:11 So i don't need to worry about the negative 4.

02:17 4 is the other side of the equation.

02:20 I still have this two -thirds exponent.

02:23 I'm going to go ahead and do the same thing...

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