5.5-Square Roots
5.5-Square Roots Important Formulae
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Grade 8 → Math → Squares and Square Roots → 5.5-Square Roots
5.5 - Square Roots
- The square root of a number is a value that, when multiplied by itself, gives the original number.
- The square root of a number $x$ is written as $\sqrt{x}$.
- For a number $x$, if $y^2 = x$, then $y = \sqrt{x}$.
- Perfect squares have whole number square roots (e.g., $\sqrt{16} = 4$).
- Non-perfect squares have irrational square roots (e.g., $\sqrt{2}$ is irrational).
- The square root of 0 is 0, i.e., $\sqrt{0} = 0$.
- To estimate square roots of non-perfect squares, use methods like long division or approximation.
5.5 - Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. In other words, if $x^2 = a$, then $x$ is the square root of $a$. The square root of a number is denoted as $\sqrt{a}$.
For example:
- $\sqrt{16} = 4$ because $4 \times 4 = 16$.
- $\sqrt{25} = 5$ because $5 \times 5 = 25$.
Key Points to Remember:
- The square root of a perfect square is an integer. For example, $ \sqrt{36} = 6$.
- Non-perfect square numbers do not have an integer square root, but they can have a decimal or irrational square root. For example, $ \sqrt{7} \approx 2.6457513110645906$.
- The square root of 0 is 0, i.e., $ \sqrt{0} = 0$.
- The square root of any positive number is always non-negative, i.e., $\sqrt{a} \geq 0$ where $a \geq 0$.
Perfect Squares:
A perfect square is a number that can be expressed as the product of an integer multiplied by itself. The square roots of perfect squares are integers. For example:
- $\sqrt{1} = 1$
- $\sqrt{4} = 2$
- $\sqrt{9} = 3$
- $\sqrt{16} = 4$
- $\sqrt{25} = 5$
- $\sqrt{36} = 6$
- $\sqrt{49} = 7$
- $\sqrt{64} = 8$
- $\sqrt{81} = 9$
Square Roots of Fractions:
The square root of a fraction is the square root of the numerator divided by the square root of the denominator. For example:
- $\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$.
- $\sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}$.
Square Roots of Decimals:
To find the square root of a decimal number, convert the decimal to a fraction or use a calculator for approximation. For example:
- $\sqrt{0.25} = 0.5$ because $0.5 \times 0.5 = 0.25$.
- $\sqrt{1.44} = 1.2$ because $1.2 \times 1.2 = 1.44$.
Estimation of Square Roots:
If the number is not a perfect square, the square root can be estimated between two perfect squares. For example, to estimate $ \sqrt{50}$:
- Find the perfect squares nearest to 50: $ \sqrt{49} = 7$ and $ \sqrt{64} = 8$.
- Since $50$ is between $49$ and $64$, we estimate $ \sqrt{50} \approx 7.1$ or $7.2$.
Finding Square Roots by Prime Factorization:
To find the square root of a number using prime factorization, express the number as a product of prime factors and group the factors in pairs. For example:
- Find the prime factorization of 36: $36 = 2^2 \times 3^2$.
- Group the factors: $(2 \times 3)^2 = 6^2$.
- Thus, $\sqrt{36} = 6$.