Exponential, Radical, and Rational Equations

TSI Math Study Guide

Exponential, Radical, and Rational Equations

This lesson brings together several types of equations that don’t behave like ordinary linear equations.

What’s tested: solving basic exponential and radical equations, modeling growth and decay, simplifying rational expressions, solving rational equations, and rearranging formulas to isolate a variable. The key is recognizing what kind of equation you’re looking at, then using the right method.

Exponential equations

Exponential equation — the variable is in the exponent. When both sides can be written with the same base, set the exponents equal.
2x = 16
2x = 2416 = 24
x = 4same base, so exponents are equal
3x + 1 = 273x + 1 = 3327 = 33x + 1 = 3set exponents equalx = 2
52x = 12552x = 53125 = 532x = 3x = 32

The same idea works with fractional bases and negative exponents.

Exponential growth and decay

Exponential functions model quantities that increase or decrease by a constant percentage over equal time intervals.

A = P(1 + r)tP  initial amountr  growth rate as a decimalt  number of time periodsA  final amount  ·  for decay, use (1 − r)

Growth

A town of 8,000 grows 3% per year. Population after 4 years?

A = 8000(1.03)4A ≈ 8000(1.1255)A ≈ 9,004

Decay — depreciation

A $24,000 car loses 15% of its value each year. Worth after 2 years?

A = 24000(1 − 0.15)2A = 24000(0.85)2A = 24000(0.7225)A = $17,340
⚠ Watch out

The same dollar amount is not subtracted each year. The 15% is applied to the new value each time.

Linear vs. exponential change

Linear

Changes by a constant amount. A savings account that grows by $500 each year.

Exponential

Changes by a constant percent or factor. An account that grows by 5% each year.

The formulas and graphs behave differently, so this distinction matters.

Radical equations

Radical equation — the variable is inside a square root or other root. To remove a square root, square both sides.
x + 5 = 4
(x + 5)2 = 42square both sides
x + 5 = 16
x = 11check: 11 + 5 = 16 = 4 ✓

Isolate the radical first

2x + 1 + 3 = 8
2x + 1 = 5subtract 3 before squaring
2x + 1 = 25square both sides
2x = 24
x = 12check: 2(12) + 1 + 3 = 5 + 3 = 8 ✓

Extraneous solutions

Extraneous solution — a value produced by squaring both sides that doesn’t work in the original equation.
x + 1 = x − 1
x + 1 = (x − 1)2square both sides
x + 1 = x2 − 2x + 1expand
0 = x2 − 3xmove everything to one side
0 = x(x − 3)factor
x = 0  or  x = 3possible solutions — now check each

Check x = 0

0 + 1 = 0 − 1  →  1 = −1

False. Extraneous.

Check x = 3

3 + 1 = 3 − 1  →  2 = 2

True. The only solution is x = 3.

Rational expressions

Rational expression — a fraction containing polynomials. The denominator can’t be zero, so some values are excluded from the domain.

x + 3x − 2    x − 2 ≠ 0,  so  x ≠ 2

Simplifying by factoring

x2 − 9x + 3
(x − 3)(x + 3)x + 3factor the numerator
x − 3,  x ≠ −3cancel the common factor; keep the restriction
⚠ Cancel factors, not termsx + 4x

Can’t be simplified by “canceling the x.” The numerator is a sum, not a product.

Multiplying rational expressions

3xx + 2 · x + 26cancel the common factor x + 2
3x6
x2,  x ≠ −2

Factoring before multiplying usually makes the calculation easier.

Solving rational equations

Multiply every term by the least common denominator so the fractions disappear.

x3 + x6 = 5LCD is 6
6(x3) + 6(x6) = 6(5)multiply every term by 6
2x + x = 30
x = 103x = 30

Variable in the denominator

2x = 6x + 4restrictions first: x ≠ 0, x ≠ −4
2(x + 4) = 6xcross multiply
2x + 8 = 6xdistribute
8 = 4xsubtract 2x
x = 2not an excluded value, so it’s valid

Rearranging formulas

The TSI may ask you to solve a formula for a particular variable. Treat all the other variables as constants and use inverse operations until the requested variable stands alone.

Solve A = ½bh for h

2A = bhmultiply by 2h = 2Abdivide by b

Solve C = 2πr for r

r = C2πdivide by 2π

Solve y = mx + b for x

y − b = mxsubtract bx = y − bmdivide by m

TSI strategy: identify the equation type first

Variable in an exponent

Write both sides with the same base.

Variable inside a radical

Isolate the radical, then raise both sides to the right power.

Variable in a denominator

Note the excluded values, then clear the fractions.

Solving a formula for a variable

Inverse operations until it stands alone.

Recognizing the type saves more time than memorizing a long list of procedures.

🔑 Key tip

With radical and rational equations, always check whether your solution is actually allowed. Squaring can introduce an extraneous solution; a value that makes a denominator zero is never valid.

A solution is only correct if it satisfies the original equation and doesn’t violate any restrictions.

Exponential, Radical, and Rational Equations Review Quiz

Next: Area, Perimeter, and Volume

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