91Ó°ÊÓ

A rational number is any number that can be expressed as the quotient or fraction of two integers. In simple terms, a rational number can be written in the form of \( \frac{a}{b} \), where \(a\) and \(b\) are integers and \(b eq 0\). Examples include numbers like \( \frac{1}{2} \), \( 3 \), and \( -4.75 \). If a number can be expressed without an endless stream of non-repeating digits, it is considered rational.
When it comes to square roots, if the result is a finite or repeating decimal, it is a rational number. In the case of our exercise, the square root of 0.0256 is 0.16, a finite decimal that confirms it as a rational number.
It's important to contrast rational numbers with irrational numbers. Irrational numbers cannot be expressed as a simple fraction and have non-repeating, infinite decimal representations. Think of \( \pi \) or \( \sqrt{2} \) as classic examples.

• A number is rational if it can be written as \( \frac{a}{b} \).
• Rational numbers can be whole numbers, fractions, or decimals.
• The square root of a perfect square is always rational.

A decimal number is a number expressed in the base-10 number system, characterized by the use of a decimal point. This point separates the integer part from the fractional part of the number. For instance, the number 3.14 has 3 as the integer part and 0.14 as the fractional part.
Decimal numbers can be either terminating or repeating. Terminating decimals have a finite number of digits after the decimal point. An example is 0.16, like the result from our step-by-step solution, as it ends after two decimal places.
Repeating decimals are those that go on indefinitely but exhibit a repeating pattern. It is also important to notice that every terminating and repeating decimal represents a rational number. The original number we worked with, 0.0256, is a decimal number that happens to be a perfect square, making its square root a simple terminating decimal.

• Decimals are based on ten, utilizing a decimal point to separate whole numbers from fractions.
• Terminating decimals have a limited number of digits after the decimal point.
• Repeating decimals have a recurring sequence of digits.

A perfect square is a number that can be expressed as the square of an integer. It means you can obtain it by multiplying an integer by itself. For example, 4 is a perfect square because \(2 \times 2 = 4\). However, perfect squares are not limited to whole numbers; decimals can also be perfect squares.
In the exercise provided, 0.0256 is recognized as a perfect square because it is equal to \( 0.16 \times 0.16 \). Therefore, when you find the square root of a perfect square, the result will be a rational number, like 0.16 in this case.
Understanding perfect squares is crucial in simplifying square roots and resolving equations involving squares. Recognizing whether a number is a perfect square not only helps in quick calculations but also in confirming the rationality of square roots.

• Perfect squares are numbers obtained by squaring an integer.
• Both whole numbers and decimals can be perfect squares.
• The square root of a perfect square is always a neat rational number.