Understanding the square root of 90 simplified helps you work faster with radicals in algebra and geometry. This guide breaks down the process so you can express the result in its simplest exact form and as a practical decimal.
Below is a quick reference that captures the key properties and steps for simplifying the square root of 90.
| Expression | Exact Form | Decimal (2 places) | Simplified Radical Form |
|---|---|---|---|
| Square root of 90 | √90 | 9.49 | 3√10 |
| Square of simplified result | (3√10)² | 90.00 | 90 |
| Related perfect square factor | 9 | — | Used in simplification |
| Is 90 a perfect square? | No | — | No integer square equals 90 |
Break down into prime factors
To simplify the square root of 90, start by expressing 90 as a product of its prime factors. This reveals any perfect squares hidden inside the radical.
90 can be factored into 2 × 3 × 3 × 5, or 2 × 3² × 5. Writing the number in this form makes it easy to identify which factors can be taken outside the square root.
Simplify the radical expression
Using the factorization 3² × 10, you can simplify √90 by pulling the square of 3 out of the radical. The result is 3√10, which is the simplest exact form.
Because 10 has no square factors other than 1, the expression 3√10 cannot be simplified further using integers.
Decimal approximation and use cases
The square root of 90 simplified in exact radical form is 3√10, but it is often useful to see its decimal equivalent. Calculating √90 gives approximately 9.49 when rounded to two decimal places.
This approximation is helpful in real-world contexts such as measuring the diagonal of a rectangle or estimating distances where an exact radical is less practical.
Check by squaring the result
You can verify that 3√10 is correct by squaring it. Squaring 3 gives 9, and squaring √10 gives 10, so 9 × 10 equals 90, confirming the simplification.
This consistency check ensures that no mistakes were made when extracting the perfect square factor from under the radical.
Key takeaways and recommended steps
- Factor 90 into 3² × 10 to reveal the perfect square.
- Pull the square root of 3² out of the radical, leaving 3 outside.
- Retain the remaining factor 10 inside the radical, giving 3√10.
- Use the decimal form 9.49 for practical measurements when an exact radical is unnecessary.
- Verify your work by squaring the simplified expression to ensure it equals 90.
FAQ
Reader questions
Is 90 a perfect square, and why does that matter for simplification?
No, 90 is not a perfect square because no integer multiplied by itself equals 90. This matters because only perfect square factors can be fully taken out of the square root, which is why the simplified form still contains a radical.
Can the square root of 90 be simplified to a whole number?
No, the square root of 90 cannot be simplified to a whole number. The exact simplified form is 3√10, which is an irrational number and continues as a non-repeating, non-terminating decimal.
How do you verify that 3√10 is the correct simplification of √90?
You verify it by squaring 3√10, which gives 9 × 10 = 90, matching the original number under the radical. This confirms that the simplification was performed correctly.
What are the steps to simplify the square root of 90 without a calculator?
Factor 90 into 3² × 10, identify the perfect square 3², take the 3 outside the radical, and leave 10 inside. This yields the simplified form 3√10.