Square Roots and Cube Roots

Squaring a number and taking the square root are opposite operations. The same is true for cubing a number and finding its cube root. Understanding these concepts is essential for solving algebra and geometry problems.

What is a Square Root?

The square root of a number $a$ is the nonnegative value $x$ that satisfies the equation $x^2 = a$. The symbol for a square root is $\sqrt{}$.

For example, $\sqrt{64} = 8$ because $8^2 = 64$.

To quickly solve problems, you should memorize the perfect squares up to $225$:

• $1^2 = 1 \implies \sqrt{1} = 1$
• $2^2 = 4 \implies \sqrt{4} = 2$
• $3^2 = 9 \implies \sqrt{9} = 3$
• $4^2 = 16 \implies \sqrt{16} = 4$
• $5^2 = 25 \implies \sqrt{25} = 5$
• $6^2 = 36 \implies \sqrt{36} = 6$
• $7^2 = 49 \implies \sqrt{49} = 7$
• $8^2 = 64 \implies \sqrt{64} = 8$
• $9^2 = 81 \implies \sqrt{81} = 9$
• $10^2 = 100 \implies \sqrt{100} = 10$
• $11^2 = 121 \implies \sqrt{121} = 11$
• $12^2 = 144 \implies \sqrt{144} = 12$
• $13^2 = 169 \implies \sqrt{169} = 13$
• $14^2 = 196 \implies \sqrt{196} = 14$
• $15^2 = 225 \implies \sqrt{225} = 15$

What is a Cube Root?

The cube root of a number $a$ is the value $x$ that satisfies the equation $x^3 = a$. The symbol for a cube root is $\sqrt[3]{}$.

Unlike square roots, cube roots can be negative! This is because a negative number multiplied by itself three times remains negative.

• Positive example: $\sqrt[3]{27} = 3$ because $3 \times 3 \times 3 = 27$.
• Negative example: $\sqrt[3]{-8} = -2$ because $(-2) \times (-2) \times (-2) = -8$.

Here are the perfect cubes up to $1000$ you should know:

• $1^3 = 1 \implies \sqrt[3]{1} = 1$
• $2^3 = 8 \implies \sqrt[3]{8} = 2$
• $3^3 = 27 \implies \sqrt[3]{27} = 3$
• $4^3 = 64 \implies \sqrt[3]{64} = 4$
• $5^3 = 125 \implies \sqrt[3]{125} = 5$
• $6^3 = 216 \implies \sqrt[3]{216} = 6$
• $7^3 = 343 \implies \sqrt[3]{343} = 7$
• $8^3 = 512 \implies \sqrt[3]{512} = 8$
• $9^3 = 729 \implies \sqrt[3]{729} = 9$
• $10^3 = 1000 \implies \sqrt[3]{1000} = 10$

Estimating Roots of Non-Perfect Squares

Not all numbers are perfect squares. When you need to find the square root of a number like $50$, you can estimate it by looking at the perfect squares around it.

Example: Estimate $\sqrt{50}$ to one decimal place.

1. Find the nearest perfect squares: We know that $49$ and $64$ are the closest perfect squares to $50$. Since $49 < 50 < 64$, it means that $\sqrt{49} < \sqrt{50} < \sqrt{64}$. Therefore, $7 < \sqrt{50} < 8$.

2. Determine the decimal: The number $50$ is much closer to $49$ than it is to $64$. So, the decimal will be very close to $7.0$. Let's test $7.1$.

3. Test your estimate: $7.1 \times 7.1 = 50.41$ Since $50.41$ is incredibly close to $50$, we can confidently estimate that $\sqrt{50} \approx 7.1$.