Step-by-Step Guidance: Simplifying Square Root Expressions

Q1. Write the expression $\sqrt{27}$ in simplest radical form.

Background

Topic: Simplifying Radical Expressions

This question tests your ability to simplify square roots by factoring out perfect squares.

Key Terms and Formulas

• Radical: An expression that uses a root, such as a square root ($\sqrt{\ }$).

• Perfect Square: A number that is the square of an integer (e.g., 1, 4, 9, 16, 25, ...).

• Simplest Radical Form: A radical expression where no perfect square factors remain under the radical sign.

Step-by-Step Guidance

1. Factor 27 into its prime factors: $27 = 3 \times 3 \times 3 = 3^3$.

2. Identify any perfect square factors within 27. Recall that $9 = 3^2$ is a perfect square.

3. Rewrite $\sqrt{27}$ as $\sqrt{9 \times 3}$ to separate the perfect square from the remaining factor.

4. Use the property $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ to split the radical.

Try solving on your own before revealing the answer!

Q2. Write the expression $\sqrt{45}$ in simplest radical form.

Background

Topic: Simplifying Radical Expressions

This question asks you to simplify a square root by factoring out perfect squares.

Key Terms and Formulas

• Perfect squares: 1, 4, 9, 16, 25, 36, ...

• Property: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

Step-by-Step Guidance

1. Factor 45 into its prime factors: $45 = 9 \times 5$.

2. Recognize that 9 is a perfect square.

3. Rewrite $\sqrt{45}$ as $\sqrt{9 \times 5}$.

4. Apply the property to split the radical into $\sqrt{9} \times \sqrt{5}$.

Try solving on your own before revealing the answer!

Q3. Write the expression $\sqrt{8}$ in simplest radical form.

Background

Topic: Simplifying Radical Expressions

This question focuses on breaking down the number under the radical to extract perfect squares.

Key Terms and Formulas

• Perfect squares: 1, 4, 9, 16, ...

• Property: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

Step-by-Step Guidance

1. Factor 8 into $4 \times 2$.

2. Notice that 4 is a perfect square.

3. Rewrite $\sqrt{8}$ as $\sqrt{4 \times 2}$.

4. Split the radical: $\sqrt{4} \times \sqrt{2}$.

Try solving on your own before revealing the answer!

Q4. Write the expression $\sqrt{27}$ in simplest radical form.

Background

Topic: Simplifying Radical Expressions

This is similar to Q1, but double-check if there is a coefficient or variable in the original question.

Key Terms and Formulas

• Perfect squares: 1, 4, 9, 16, ...

• Property: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

Step-by-Step Guidance

1. Factor 27 into $9 \times 3$.

2. Recognize 9 as a perfect square.

3. Rewrite $\sqrt{27}$ as $\sqrt{9 \times 3}$.

4. Split the radical: $\sqrt{9} \times \sqrt{3}$.

Try solving on your own before revealing the answer!

Q5. Write the expression $\sqrt{108}$ in simplest radical form.

Background

Topic: Simplifying Radical Expressions

This question involves simplifying a larger number under the radical by factoring out perfect squares.

Key Terms and Formulas

• Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...

• Property: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

Step-by-Step Guidance

1. Factor 108 into its prime factors: $108 = 2 \times 2 \times 3 \times 3 \times 3$.

2. Group the factors to find the largest perfect square. $108 = 36 \times 3$.

3. Rewrite $\sqrt{108}$ as $\sqrt{36 \times 3}$.

4. Split the radical: $\sqrt{36} \times \sqrt{3}$.

Try solving on your own before revealing the answer!

Q6. Write the expression $\sqrt{245}$ in simplest radical form.

Background

Topic: Simplifying Radical Expressions

This question asks you to factor a larger number to extract perfect squares from under the radical.

Key Terms and Formulas

• Perfect squares: 1, 4, 9, 16, 25, 36, 49, ...

• Property: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

Step-by-Step Guidance

1. Factor 245 into $49 \times 5$.

2. Recognize 49 as a perfect square.

3. Rewrite $\sqrt{245}$ as $\sqrt{49 \times 5}$.

4. Split the radical: $\sqrt{49} \times \sqrt{5}$.

Try solving on your own before revealing the answer!

Q7. Write the expression $\sqrt{180}$ in simplest radical form.

Background

Topic: Simplifying Radical Expressions

This question involves simplifying a square root by factoring out the largest perfect square.

Key Terms and Formulas

• Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ...

• Property: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

Step-by-Step Guidance

1. Factor 180 into $36 \times 5$.

2. Recognize 36 as a perfect square.

3. Rewrite $\sqrt{180}$ as $\sqrt{36 \times 5}$.

4. Split the radical: $\sqrt{36} \times \sqrt{5}$.

Try solving on your own before revealing the answer!

Q8. Write the expression $\sqrt{125}$ in simplest radical form.

Background

Topic: Simplifying Radical Expressions

This question asks you to simplify a square root by factoring out perfect squares.

Key Terms and Formulas

• Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, ...

• Property: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

Step-by-Step Guidance

1. Factor 125 into $25 \times 5$.

2. Recognize 25 as a perfect square.

3. Rewrite $\sqrt{125}$ as $\sqrt{25 \times 5}$.

4. Split the radical: $\sqrt{25} \times \sqrt{5}$.

Try solving on your own before revealing the answer!

Q9. Select all expressions that are equivalent to $\sqrt{75}$.

Background

Topic: Equivalent Radical Expressions

This question tests your understanding of how to rewrite and recognize equivalent forms of a radical expression.

Key Terms and Formulas

• Equivalent expressions: Different expressions that simplify to the same value.

• Property: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

• Property: $\sqrt{a^2} = a$ for $a \geq 0$

Step-by-Step Guidance

1. Factor 75 into $25 \times 3$ to find its simplest radical form.

2. Rewrite $\sqrt{75}$ as $\sqrt{25} \times \sqrt{3}$.

3. Recall that $\sqrt{25} = 5$, so $\sqrt{75} = 5\sqrt{3}$.

4. Compare this form to the given answer choices to determine which are equivalent.

Try solving on your own before revealing the answer!

Q10. Select all expressions that are equivalent to $\sqrt{75}$.

Background

Topic: Equivalent Radical Expressions

This question is similar to Q9 and asks you to identify all expressions that are equivalent to $\sqrt{75}$.

Key Terms and Formulas

• Equivalent expressions: Different expressions that simplify to the same value.

• Property: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

Step-by-Step Guidance

1. Recall from the previous question that $\sqrt{75} = 5\sqrt{3}$.

2. Check each given expression by simplifying it to see if it matches $5\sqrt{3}$.

3. For example, $\sqrt{25} \times \sqrt{3} = 5\sqrt{3}$, and $\sqrt{9 \times 25} = \sqrt{225} = 15$.

4. Continue this process for each option to determine equivalence.

Try solving on your own before revealing the answer!