3 Ways To Express √90 In Radical Form

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The concept of expressing a number in radical form is a fundamental idea in mathematics, particularly in algebra and geometry. It involves representing a number as a product of its prime factors, where one of the factors is a square root. In this article, we will explore three different ways to express √90 in radical form.

Understanding Radical Form

Before we dive into expressing √90 in radical form, let's first understand what radical form means. Radical form is a way of representing a number as a product of its prime factors, where one of the factors is a square root. This form is particularly useful when working with numbers that are not perfect squares.

What is a Perfect Square?

A perfect square is a number that can be expressed as the product of an integer and itself. For example, 16 is a perfect square because it can be expressed as 4 × 4. On the other hand, 90 is not a perfect square because it cannot be expressed as the product of an integer and itself.

Method 1: Prime Factorization

One way to express √90 in radical form is by using prime factorization. Prime factorization involves breaking down a number into its prime factors. The prime factors of 90 are 2, 3, 3, and 5. We can express √90 as:

√90 = √(2 × 3 × 3 × 5)

Using the properties of radicals, we can simplify this expression as:

√90 = √(9 × 10)

= √9 × √10

= 3√10

Method 1: Simplifying Radical Expressions

Simplifying radical expressions involves breaking down a radical expression into simpler components. In this case, we simplified the expression √90 by breaking it down into √9 and √10.

Method 2: Using the Conjugate Pair

Another way to express √90 in radical form is by using the conjugate pair. The conjugate pair of √90 is √90 and -√90. We can express √90 as:

√90 = (√90)(√90 + √90) / (√90 + √90)

Using the conjugate pair, we can simplify this expression as:

√90 = (√90)(√90 + √90) / 2√90

= (√90)(√90 + √90) / 2√90

= (√90 + √90) / 2

= 3√10 / 2

Method 2: Rationalizing the Denominator

Rationalizing the denominator involves eliminating any radicals in the denominator. In this case, we rationalized the denominator by multiplying both the numerator and denominator by the conjugate pair.

Method 3: Using the Square Root Property

A third way to express √90 in radical form is by using the square root property. The square root property states that if a^2 = b, then a = √b. We can express √90 as:

√90 = √(3^2 × 10)

Using the square root property, we can simplify this expression as:

√90 = 3√10

Method 3: Simplifying Radical Expressions

Simplifying radical expressions involves breaking down a radical expression into simpler components. In this case, we simplified the expression √90 by breaking it down into √9 and √10.

Conclusion

In this article, we explored three different ways to express √90 in radical form. We used prime factorization, the conjugate pair, and the square root property to simplify the expression. Each method provides a unique way of expressing √90 in radical form.