Name: Decimals Word Problems And Worked Solutions
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The square root of a number x is a value y such that when multiplied by itself, it produces the original number:

\sqrt{x} = y \iff y^2 = x \quad (\text{where } y \ge 0)

Calculating the square root of a decimal number without a calculator relies on converting the decimal into a fraction or using power-of-ten index laws.

1. The Fraction Conversion Method

The most reliable non-calculator method for finding the square root of a terminating decimal is to write it as an equivalent common fraction whose denominator is a power of 10:

\sqrt{x} = \sqrt{\dfrac{a}{10^{2k}}} = \dfrac{\sqrt{a}}{\sqrt{10^{2k}}} = \dfrac{\sqrt{a}}{10^k}

Because the square root of a quotient equals the quotient of the square roots:

\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}

Convert to a fraction: Count the decimal places. If there are 2 decimal places, write over 100; if 4 decimal places, write over 10,000; if 6 decimal places, write over 1,000,000.
Take the square root of the numerator and denominator independently.
Convert back to a decimal by dividing by the power of 10.

2. The Decimal Place Halving Rule

When squaring a decimal, the number of decimal places doubles:

(0.3)^2 = 0.3 \times 0.3 = 0.09 \quad (1\text{ d.p.} \to 2\text{ d.p.})

(0.12)^2 = 0.12 \times 0.12 = 0.0144 \quad (2\text{ d.p.} \to 4\text{ d.p.})

Conversely, when taking the square root of a perfect square decimal:

\text{Decimal places in } \sqrt{x} = \dfrac{\text{Decimal places in } x}{2}

The Even Decimal Place Rule:

A decimal can only have an exact rational terminating square root if it has an even number of decimal places (2, 4, 6, etc.) and its integer digits form a perfect square.

3. Odd Decimal Places and Surd Representation

If a decimal has an odd number of decimal places, its square root is an irrational surd rather than a terminating decimal:

\sqrt{0.4} = \sqrt{\dfrac{4}{10}} = \dfrac{\sqrt{4}}{\sqrt{10}} = \dfrac{2}{\sqrt{10}} = \dfrac{2\sqrt{10}}{10} = \dfrac{\sqrt{10}}{5} \approx 0.6325

Notice that $\sqrt{0.4} \neq 0.$ 2, because $(0.2)^2 = 0.04$ , not $0.4$ .

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Practice Questions & Solutions

Score:

Calculate the exact value of:

(a) $\text{[math]}$

(b) $\text{[math]}$

Solution

(a) Evaluate $\text{[math]}$

Step 1: Convert the decimal to a fraction with a denominator of 100

$\text{[math]}$

Step 2: Apply the square root to numerator and denominator

$\text{[math]}$

Step 3: Convert back to decimal form

$\text{[math]}$

(b) Evaluate $\text{[math]}$

Step 1: Convert the decimal to a fraction with a denominator of 10,000

$\text{[math]}$

Step 2: Take the square root of numerator and denominator

$\text{[math]}$

Step 3: Convert back to decimal form

$\text{[math]}$

Evaluate the square roots:

(a) $\text{[math]}$

(b) $\text{[math]}$

Solution

(a) Evaluate $\text{[math]}$

Step 1: Convert to an improper fraction

$\text{[math]}$

Step 2: Take the square root of numerator and denominator

$\text{[math]}$

Step 3: Convert back to decimal form

$\text{[math]}$

(b) Evaluate $\text{[math]}$

Step 1: Convert to an improper fraction

$\text{[math]}$

Step 2: Take the square root of numerator and denominator

$\text{[math]}$

Step 3: Convert back to decimal form

$\text{[math]}$

Find the exact values of:

(a) $\text{[math]}$

(b) $\text{[math]}$

Solution

(a) Evaluate $\text{[math]}$

Step 1: Convert to a fraction with 4 decimal places

$\text{[math]}$

Step 2: Apply the square root

$\text{[math]}$

Step 3: Convert to decimal form

$\text{[math]}$

(b) Evaluate $\text{[math]}$

Step 1: Convert to a fraction with 6 decimal places

$\text{[math]}$

Step 2: Apply the square root

$\text{[math]}$

Step 3: Convert to decimal form

$\text{[math]}$

Evaluate the numerical expression:

$\text{[math]}$

Solution

Evaluate $\text{[math]}$

Step 1: Evaluate each individual square root

$\text{[math]}$

Step 2: Substitute the evaluated roots into the expression

$\text{[math]}$

Step 3: Carry out multiplications

$\text{[math]}$

Step 4: Combine the terms

$\text{[math]}$

Write $\text{[math]}$ in the simplified surd form $\text{[math]}$ , where a and b are integers.

Solution

Simplify $\text{[math]}$ into surd form

Step 1: Write as a fraction

$\text{[math]}$

Step 2: Distribute the square root

$\text{[math]}$

Step 3: Rationalise the denominator by multiplying by $\text{[math]}$

$\text{[math]}$

A square tile has an area of $\text{[math]}$ . Calculate the perimeter of the tile.

Solution

Step 1: Find the side length s of the square

The area of a square is $\text{[math]}$ :

$\text{[math]}$

Since $\text{[math]}$ :

$\text{[math]}$

Step 2: Calculate the perimeter

$\text{[math]}$

Under uniform acceleration from rest, final speed is given by $\text{[math]}$ .

Calculate the exact value of $\text{[math]}$ when $\text{[math]}$ and $\text{[math]}$ .

Solution

Step 1: Calculate the value inside the radical

$\text{[math]}$

Step 2: Evaluate the square root of 7.84

$\text{[math]}$

Since $\text{[math]}$ :

$\text{[math]}$

The area of one square face of a solid cube is $\text{[math]}$ .

(a) Find the edge length of the cube in metres.

(b) Calculate the total surface area across all six faces of the cube.

Solution

(a) Find the Edge Length of the Cube
The area of a square face is given by $\text{[math]}$ , where s represents the edge length.

Step 1: Convert the decimal into a fraction with a denominator of 10,000

$\text{[math]}$

Step 2: Apply the square root to numerator and denominator

$\text{[math]}$

Since $\text{[math]}$ and $\text{[math]}$ :

$\text{[math]}$

The edge length of the cube is $\text{[math]}$ .

(b) Calculate the Total Surface Area
A cube consists of six identical square faces:

$\text{[math]}$

Substitute the face area $\text{[math]}$ :

$\text{[math]}$

The total surface area is $\text{[math]}$ .

$\text{[math]}$

Calculate using fractions or powers of 15:

$\text{[math]}$

Convert into standard scientific form $\text{[math]}$ , where $\text{[math]}$ :

Move the decimal point 3 places to the right:

$\text{[math]}$

Evaluate the exact value of:

$\text{[math]}$

Solution

Evaluate $\text{[math]}$

Step 1: Simplify the decimal fraction inside the square root

Multiply numerator and denominator by 10 to clear the denominator's decimal:

$\text{[math]}$

Step 2: Take the square root of 0.36

$\text{[math]}$

Evaluate the following expression, giving your answer in standard scientific form $\text{[math]}$ :

$\text{[math]}$

Solution

Evaluate $\text{[math]}$

Step 1: Separate the numerical factor and the power of ten

$\text{[math]}$

Step 2: Evaluate the square root of each component

$\text{[math]}$

Using index laws $\text{[math]}$

Step 3: Combine in standard scientific notation

$\text{[math]}$

(In standard decimal notation: $\text{[math]}$ ).

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Gianpiero Placidi

UK-based Chemistry graduate with a passion for education, providing clear explanations and thoughtful guidance to inspire student success.

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shabana sivasankar

January 2026

what will be the answer when 12.345 is subtracted from 234.00 multiplied by 778.90 and divided by67.23

Vanessa - Editorial Manager Superprof UK

January 2026

If we follow the order of operations (multiply and divide first, then subtract):

1. (234.00 \times 778.90 = 182,262.6)
2. (182,262.6 \div 67.23 \approx 2711.03)
3. (2711.03 – 12.345 \approx 2698.69)

**Final answer: ≈ 2,698.69**

Diti

September 2024

If it is (0.0072÷6)

Ashutosh Gond

June 2024

find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081
find the square root in decimal form of 6.7081

Ashutosh Gond

June 2024

find the square root in decimal form of 6.7081

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Chapters

Practice Questions & Solutions

Square Rooting Decimals: Practice Problems and Solutions

1. The Fraction Conversion Method

2. The Decimal Place Halving Rule

3. Odd Decimal Places and Surd Representation

Practice Questions & Solutions

Adding and Subtracting Decimal Numbers

Dividing Decimal Numbers

Multiplying Decimal Numbers

Ordering Decimals

What Are The Different Types of Decimals?

What is A Decimal Number? Place Value Made Simple

How to Find the Square Root of a Decimal

Decimal Number Worksheets

Decimals Word Problems And Worked Solutions

Cancel reply

Square Rooting Decimals: Practice Problems and Solutions

1. The Fraction Conversion Method

2. The Decimal Place Halving Rule

3. Odd Decimal Places and Surd Representation

Practice Questions & Solutions

Theory

Adding and Subtracting Decimal Numbers

Dividing Decimal Numbers

Multiplying Decimal Numbers

Ordering Decimals

What Are The Different Types of Decimals?

What is A Decimal Number? Place Value Made Simple

How to Find the Square Root of a Decimal

Exercises

Decimal Number Worksheets

Decimals Word Problems And Worked Solutions

Cancel reply