A perfect square is a number that can be expressed as the product of two equal integers. In other words, it is a number that can be represented as the square of a whole number.
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The square root of a perfect square is always a whole number.
Perfect squares are important in the context of simplifying square roots and multiplying square roots.
Identifying perfect squares within a radical expression can help simplify the expression.
The product of two perfect squares is also a perfect square.
Perfect squares can be used to rationalize the denominator of a fraction with a square root in the denominator.
Review Questions
Explain how the concept of perfect squares relates to simplifying square root expressions.
When simplifying square root expressions, it is important to identify any perfect squares within the radicand. This is because the square root of a perfect square is a whole number, which can then be taken outside the radical sign, simplifying the expression. For example, the square root of 36 can be simplified to 6, since 36 is a perfect square (6^2 = 36).
Describe how the properties of perfect squares can be used to multiply square root expressions.
When multiplying square root expressions, the property of perfect squares can be used to simplify the calculation. If the radicands of the two square root expressions are perfect squares, then the product of the square roots is also a perfect square. For instance, $\sqrt{9} \times \sqrt{16} = \sqrt{9 \times 16} = \sqrt{144} = 12$, since 9 and 16 are both perfect squares.
Analyze how the concept of perfect squares can be used to rationalize the denominator of a fraction with a square root in the denominator.
To rationalize the denominator of a fraction with a square root, the goal is to eliminate the square root from the denominator. This can be achieved by multiplying both the numerator and denominator by the conjugate of the denominator, which is the square root expression with the opposite sign. If the denominator contains a perfect square, this process is simplified, as the product of the square root and its conjugate will result in the perfect square, removing the radical from the denominator. For example, $\frac{1}{\sqrt{25}} = \frac{\sqrt{25}}{25} = \frac{5}{25} = \frac{1}{5}$, since 25 is a perfect square.