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 constantly when solving equations, so getting comfortable simplifying and manipulating them is essential for the rest of 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.