Solve Radical Equations

Lesson

Practice

Lesson

Radical terms are terms that have a root sign, such as a square root or cube root. Some equations contain radical terms and, in this lesson, we are going to look at how to solve these kinds of equations.

The process is basically the same as solving "regular" linear equations. We can still use backtracking (that is, using the opposite operation) to solve equations. So how do we backtrack radical terms? We use exponents or powers.

Proof:

The fractional index law states: $x^{\frac{1}{m}}=\sqrt[m]{x}$x1m=^m√x

The power of a power law states: $\left(x^m\right)^n=x^{mn}$(xm)n=xmn

So:

$\left(\sqrt[m]{x}\right)^m$(^m√`x`)`m`	$=$=	$\left(x^{\frac{1}{m}}\right)^m$(`x`1`m`)`m`
	$=$=	$x^{\frac{m}{m}}$`xmm`
	$=$=	$x^1$`x`1
	$=$=	$x$`x`

Worked Examples

QUESTION 1

Solve $\sqrt{x}=\sqrt{13}$√x=√13.

QUESTION 2

$\sqrt{m}+2=0$√m+2=0

Solve for $m$m.
Find the value of $\sqrt{m}+2$√m+2 when $m=4$m=4.
Is $m=4$m=4 a solution of $\sqrt{m}+2=0$√m+2=0?
Yes
A
No
B

QUESTION 3

Solve $\sqrt{-5x+2}=\sqrt{-2x+8}$√−5x+2=√−2x+8.

Outcomes

9.A.P.2

State and motivate the Remainder Theorem with examples and analogy to integers. Statement and proof of the Factor Theorem. Factorisation of ax^2 + bx + c, a ≠ 0 where a, b, c are real numbers, and of cubic polynomials using the Factor Theorem.

9.A.P.3

Recall of algebraic expressions and identities. Further identities of the type: (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx, (x ± y)^ 3 = x^3 ± y^3 ± 3xy (x ± y), x^3 + y^3 + z^3 – 3xyz = (x + y + z) (x^2 + y^2 + z^2 – xy – yz – zx) and their use in factorization of polynomials. Simple expressions reducible to these polynomials.