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A calculator's $$\sqrt{x}$$ key can be used to find decimal approximations of square roots. For **perfect squares**, the calculator returns an **exact** value — for instance, $$\sqrt{4} = 2$$ and $$\sqrt{9} = 3$$. For numbers that are **not** perfect squares, however, the calculator display shows only an **approximation**, not the exact square root. The display is limited by the number of digits it can show.

For example, a 10-digit calculator displays $$\sqrt{5} \approx 2.236067978$$. To confirm this is an approximation and not the exact value, square the result: $$(2.236067978)^2 = 5.000000002$$, which is close to $$5$$ but not exactly $$5$$. Similarly, rounding to two decimal places gives $$\sqrt{5} \approx 2.24$$, and $$(2.24)^2 = 5.0176$$ — again close, but not equal to $$5$$.

Using the calculator and rounding to two decimal places, the square roots of $$4$$ through $$9$$ are:

| Expression | Value |
|---|---|
| $$\sqrt{4}$$ | $$= 2$$ (exact) |
| $$\sqrt{5}$$ | $$\approx 2.24$$ |
| $$\sqrt{6}$$ | $$\approx 2.45$$ |
| $$\sqrt{7}$$ | $$\approx 2.65$$ |
| $$\sqrt{8}$$ | $$\approx 2.83$$ |
| $$\sqrt{9}$$ | $$= 3$$ (exact) |

Notice that $$\sqrt{4}$$ and $$\sqrt{9}$$ use the equal sign because $$4$$ and $$9$$ are perfect squares, while the others use the approximation symbol $$\approx$$ because $$5$$, $$6$$, $$7$$, and $$8$$ are not perfect squares.

Approximating Square Roots with a Calculator

Imagine you are a quality control inspector documenting the dimensions of square components for a manufacturing report. When using a calculator to find the square root of a number, explain the fundamental difference in how the calculator handles 'perfect squares' versus 'non-perfect squares.' Additionally, describe the specific mathematical step you can take to prove that a calculator's result for a non-perfect square is an approximation rather than an exact value.

Distinguishing Exact and Approximate Square Roots

Just as a calculator's $$\sqrt{x}$$ key provides decimal approximations for square roots, the $$\sqrt[y]{x}$$ key (or its equivalent for higher indices) is used to find decimal approximations for roots with an index greater than $$2$$. For a number that is not a perfect $$n$$th power, the calculator returns a rounded decimal approximation rather than an exact value. For example, using a calculator to evaluate $$\sqrt[4]{93}$$ yields an approximation such as $$3.105422799$$, which can be rounded to two decimal places as $$3.11$$. To verify that this is an approximation, raising the rounded result to the fourth power gives $$(3.11)^4 \approx 93.54951841$$, which is close to $$93$$ but not exactly $$93$$. The approximation symbol $$\approx$$ indicates that the value is rounded.

Approximating Higher Roots with a Calculator

A calculator display of $$\sqrt{8}$$ is an exact value.

What does $$\approx$$ mean in $$\sqrt{7} \approx 2.65$$?

Why is $$\sqrt{10} \approx 3.16227766$$ an approximation?

Order the steps to approximate $$\sqrt{7}$$ to two decimal places and check it.

A calculator shows $$\sqrt{6}\approx2.44948974$$. How can you check that this value is an approximation?

Case context:
A calculator shows $$\sqrt{9}=3$$ and $$\sqrt{5}\approx2.24$$.

Question:
Classify each calculator result as exact or approximate. Then explain why the second result is approximate.

Sample answer:
$$\sqrt{9}=3$$ is exact because $$9$$ is a perfect square. $$\sqrt{5}\approx2.24$$ is approximate because $$5$$ is not a perfect square and the calculator displays a limited, rounded number of digits. Also, $$(2.24)^2=5.0176$$, not exactly $$5$$.

Key points:
- $$\sqrt{9}=3$$ is exact.
- $$9$$ is a perfect square.
- $$\sqrt{5}\approx2.24$$ is approximate.
- $$5$$ is not a perfect square.
- The displayed decimal is limited or rounded, and its square is not exactly $$5$$.

Rubric:
The answer must classify both results correctly and give two valid reasons why $$2.24$$ is an approximation of $$\sqrt{5}$$.

Classify each calculator result as exact or approximate. Then explain why the second result is approximate.

Question:
A calculator shows $$\sqrt{11} \approx 3.31662479$$. What term describes $$3.31662479$$?

Sample answer:
It is an approximation of $$\sqrt{11}$$.

Key points:
- The result is an approximation.
- It is not the exact value of $$\sqrt{11}$$.

Feedback:
The calculator display is an approximation because $$11$$ is not a perfect square.

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