9.1 Simplify and Use Square Roots

Square roots let you find a value that, when multiplied by itself, gives a specific number. They show up often when solving equations, so simplifying and manipulating square roots is a skill you will reuse throughout algebra.

Simplifying and Using Square Roots

Simplification of square root expressions

The core idea behind simplifying a square root is pulling out perfect square factors from under the radical sign. If you can spot a perfect square hiding inside the radicand, you can simplify.

Simplifying with numbers:

1. Factor the radicand into a perfect square times whatever is left over.
2. Take the square root of the perfect square and move it outside the radical.

$\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}$

It helps to know your perfect squares cold: 4, 9, 16, 25, 36, 49, 64, 81, 100. When you look at 48, you want the largest perfect square that divides it evenly. Here, 16 goes into 48, so you use that.

Simplifying with variables:

Variables raised to even powers are perfect squares. For instance, $x^2$ is a perfect square because $\sqrt{x^2} = x$ (assuming $x \geq 0$). Similarly, $y^4$ is a perfect square because $\sqrt{y^4} = y^2$.

$\sqrt{18x^2} = \sqrt{9x^2 \cdot 2} = \sqrt{9x^2} \cdot \sqrt{2} = 3x\sqrt{2}$

Combining like terms under a radical:

If you have terms under the same square root that can be combined first, do that before simplifying.

$\sqrt{8x^2 + 18x^2} = \sqrt{26x^2} = x\sqrt{26}$

Rationalizing denominators:

A simplified expression shouldn't have a square root in the denominator. To fix this, multiply the top and bottom by the square root in the denominator:

$\frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

You're really just multiplying by 1 in a clever form, so the value doesn't change.

Estimation of square root values

When you need to estimate a square root without a calculator, find the two consecutive perfect squares it falls between.

For $\sqrt{10}$:

• $\sqrt{9} = 3$ and $\sqrt{16} = 4$, so $\sqrt{10}$ is between 3 and 4.
• Since 10 is much closer to 9 than to 16, the answer is just a little above 3. A good estimate is about 3.16.

This technique works for any square root. For $\sqrt{50}$: you know $\sqrt{49} = 7$ and $\sqrt{64} = 8$, and 50 is very close to 49, so $\sqrt{50} \approx 7.07$.

Application of square root properties

Two properties make working with square roots much easier:

Product Property: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$

You can combine two square roots being multiplied into one:

$\sqrt{3} \cdot \sqrt{12} = \sqrt{3 \cdot 12} = \sqrt{36} = 6$

Quotient Property: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

You can combine a division of square roots into one:

$\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3$

Solving equations with square roots:

1. Isolate the square root term on one side of the equation.
2. Square both sides to eliminate the square root.
3. Solve the resulting equation for the variable.

$\sqrt{x + 1} = 3$

• Square both sides: $(\sqrt{x + 1})^2 = 3^2$
• Simplify: $x + 1 = 9$
• Solve: $x = 8$

Always check your answer by plugging it back in: $\sqrt{8 + 1} = \sqrt{9} = 3$ ✓. Squaring both sides can sometimes introduce answers that don't actually work (called extraneous solutions), so checking is a good habit.

Radical vs exponential forms

Square roots can be written two ways, and you should be able to switch between them.

• Radical form uses the square root symbol: $\sqrt{a}$
• Exponential form uses a fractional exponent: $a^{\frac{1}{2}}$

These mean the exact same thing. To convert:

• Radical to exponential: $\sqrt{5} = 5^{\frac{1}{2}}$
• Exponential to radical: $7^{\frac{1}{2}} = \sqrt{7}$

The exponential form becomes especially useful later when you need to apply exponent rules to expressions involving roots.

Number Systems and Square Roots

• Square roots of perfect squares (like $\sqrt{25} = 5$) are rational numbers because they can be written as fractions of integers.
• Square roots of non-perfect squares (like $\sqrt{2}$ or $\sqrt{10}$) are irrational numbers. Their decimal forms go on forever without repeating.
• Both rational and irrational numbers are part of the real number system.
• You can add, subtract, multiply, and divide square roots, but you can only add or subtract them when they have the same radicand (just like combining like terms). For example, $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$, but $\sqrt{2} + \sqrt{3}$ cannot be simplified further.