📘Intermediate Algebra Unit 8 Review

Roots let you reverse the operation of raising a number to a power. If squaring 5 gives you 25, then taking the square root of 25 gives you back 5. This section covers how to simplify root expressions, estimate irrational roots, and work with radicals in different operations.

Square Roots

The symbol $\sqrt{x}$ means "the non-negative number that, when multiplied by itself, equals $x$ ." This non-negative result is called the principal square root.

$\sqrt{25} = 5$ because $5 \times 5 = 25$
$\sqrt{0} = 0$
$\sqrt{-9}$ is not a real number, because no real number times itself gives a negative result

Perfect squares are numbers whose square roots come out whole: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on. Memorizing at least the first 12 or so will speed up your work considerably.

To simplify a square root that isn't a perfect square, factor out the largest perfect square you can find:

Find the largest perfect square factor of the radicand (the number under the radical).
Rewrite using the product rule: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$
Simplify the perfect square part.

Example: Simplify $\sqrt{72}$

The largest perfect square factor of 72 is 36 (since $72 = 36 \times 2$ ).
$\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2}$
$= 6\sqrt{2}$

Higher-Order Roots

The expression $\sqrt[n]{x}$ means "the number that, when raised to the $n$ th power, equals $x$ ." The number $n$ is called the index of the radical.

$\sqrt[3]{8} = 2$ because $2^3 = 8$
$\sqrt[4]{81} = 3$ because $3^4 = 81$

There's an important distinction based on whether the index is even or odd:

Odd index (cube roots, 5th roots, etc.): You can take the root of negative numbers. For example, $\sqrt[3]{-27} = -3$ because $(-3)^3 = -27$ .
Even index (square roots, 4th roots, etc.): The radicand must be non-negative for a real number result. $\sqrt[4]{-16}$ is not a real number.

The simplification process works the same way as with square roots. Factor out perfect $n$ th powers:

Example: Simplify $\sqrt[3]{54}$

The largest perfect cube factor of 54 is 27 (since $54 = 27 \times 2$ ).
$\sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = \sqrt[3]{27} \cdot \sqrt[3]{2}$
$= 3\sqrt[3]{2}$

Fractional exponent connection: Any root can be rewritten as a fractional exponent: $\sqrt[n]{x} = x^{1/n}$ . This is useful when you need to apply exponent rules to radical expressions.

Simplification of root expressions, OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Simplifying Radical Expressions

Estimation of Irrational Roots

When a radicand isn't a perfect power, the result is irrational. You can estimate its value by finding which two whole numbers it falls between.

Identify the two consecutive perfect squares (or perfect $n$ th powers) that the radicand sits between.
Determine which one it's closer to.
Use a calculator for a decimal approximation if needed.

Example: Estimate $\sqrt{40}$

$\sqrt{36} = 6$ and $\sqrt{49} = 7$ , so $\sqrt{40}$ is between 6 and 7.

Since 40 is closer to 36 than to 49, the answer is closer to 6.

Calculator: $\sqrt{40} \approx 6.32$

Simplifying Radicals with Variables

The same factoring approach applies when variables appear under the radical. The key rule:

$\sqrt[n]{x^n} = |x|$ when $n$ is even, and $\sqrt[n]{x^n} = x$ when $n$ is odd.

The absolute value is needed for even indices because the principal root must be non-negative, and you don't know whether the variable is positive or negative.

Example: Simplify $\sqrt{50x^4y^3}$

Factor into perfect squares and leftovers: $\sqrt{25 \cdot 2 \cdot x^4 \cdot y^2 \cdot y}$
Take the square root of the perfect square factors: $5x^2 y\sqrt{2y}$

(Here $x^2$ is already non-negative regardless of $x$ , so no absolute value is needed for it. If the problem had $\sqrt{x^2}$ , you'd write $|x|$ .)

Simplification of root expressions, OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Radicals Including Square and Cube ...

Combining, Multiplying, and Dividing Radicals

Combining like radicals works just like combining like terms. Radicals are "like" when they share the same index and the same radicand.

$5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}$
$\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$ (simplify each radical first)

Multiplying radicals with the same index: multiply the radicands, then simplify.

$\sqrt{6} \cdot \sqrt{10} = \sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15}$

Dividing radicals with the same index: use the quotient rule $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ , then simplify.

$\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25} = 5$

Rationalizing Denominators

A simplified radical expression should have no radicals in the denominator. To eliminate them, multiply the numerator and denominator by a value that makes the denominator rational.

For a single radical in the denominator:

Multiply top and bottom by that same radical.
Simplify.

Example: $\frac{3}{\sqrt{5}} = \frac{3}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5}$

For a higher-index radical, multiply by whatever makes the radicand a perfect power:

Example: $\frac{1}{\sqrt[3]{4}} = \frac{1}{\sqrt[3]{4}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{2}} = \frac{\sqrt[3]{2}}{\sqrt[3]{8}} = \frac{\sqrt[3]{2}}{2}$

Number Systems and Roots

Rational numbers can be written as a fraction of two integers: $\frac{1}{2}$ , $-3$ , $0.75$
Irrational numbers cannot be written as such a fraction. Their decimal forms go on forever without repeating: $\sqrt{2}$ , $\pi$ , $\sqrt[3]{5}$
Real numbers include both rational and irrational numbers.

Radicals don't always produce irrational numbers. When the radicand is a perfect power, the result is rational: $\sqrt{49} = 7$ , $\sqrt[3]{64} = 4$ . A quick way to tell: if you can't simplify the radical down to a whole number, the result is irrational.

📘Intermediate Algebra Unit 8 Review