📝 Simplify Expressions with Square Roots (28 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 28 questions available

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What is Simplify Expressions with Square Roots?

Definition:
Simplifying expressions with square roots involves rewriting the square root of a number (or variable) in its simplest radical form by factoring out perfect squares from inside the radical, using the property $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ for non-negative $a, b$ , and combining like radicals.

Working:
Factor the radicand (number under root) into a product of a perfect square and another factor, then take the square root of the perfect square outside the radical; for variables, use $\sqrt{x^2} = |x|$ ; combine terms with the same radical part.

Example:
Simplify $\sqrt{50} + \sqrt{18}$ .
Solution: $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$ ; $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$ ; sum $5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$ .

Reason:
Simplifying square roots is important for solving quadratic equations, working with the Pythagorean theorem, and handling geometric problems, making expressions more manageable and easier to compare.

Easy

Medium

Hard

📝 All Simplify Expressions with Square Roots MCQs

Q1. A student simplifies $3\sqrt{12}-2\sqrt{27}$ by treating the radicands as ordinary coefficients. Which simplified expression is correct?

A.0

\sqrt{3}

✅

6\sqrt{3}

12\sqrt{3}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: First simplify each radical: $\sqrt{12}=2\sqrt{3}$ and $\sqrt{27}=3\sqrt{3}$ . Therefore $3(2\sqrt{3})-2(3\sqrt{3})=6\sqrt{3}-6\sqrt{3}=0$ . This shows why radical terms must be simplified before combining like radicals.

Q2. Which expression is equivalent to $2\sqrt{18}+\sqrt{50}-3\sqrt{8}$ ?

5\sqrt{2}

4\sqrt{2}

✅

6\sqrt{2}

8\sqrt{2}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Rewrite the radicals using perfect-square factors: $\sqrt{18}=3\sqrt{2}$ , $\sqrt{50}=5\sqrt{2}$ , and $\sqrt{8}=2\sqrt{2}$ . The expression becomes $6\sqrt{2}+5\sqrt{2}-6\sqrt{2}=5\sqrt{2}$ , so the correct choice is B.

Q3. A rectangular garden has side lengths $\sqrt{48}$ meters and $\sqrt{12}$ meters. If its perimeter is simplified exactly, which expression represents the perimeter?

10\sqrt{3}

✅

12\sqrt{3}

16\sqrt{3}

20\sqrt{3}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The perimeter is $2(\sqrt{48}+\sqrt{12})$ . Since $\sqrt{48}=4\sqrt{3}$ and $\sqrt{12}=2\sqrt{3}$ , the perimeter becomes $2(6\sqrt{3})=12\sqrt{3}$ . Thus option B is correct.

Q4. A student claims $\sqrt{72}-\sqrt{32}=\sqrt{40}$ because the radicands can be subtracted directly. What is the correct simplified result, and why is the student's method invalid?

\sqrt{40}

, because subtraction is allowed inside a radical

2\sqrt{2}

, because both radicals simplify to like radical terms ✅

4\sqrt{2}

, because the square roots cancel

10\sqrt{2}

, because the radicands add after simplification

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Radicals cannot generally be combined by subtracting their radicands. Instead, $\sqrt{72}=6\sqrt{2}$ and $\sqrt{32}=4\sqrt{2}$ . Subtracting these like radical terms gives $6\sqrt{2}-4\sqrt{2}=2\sqrt{2}$ .

Q5. A graph shows a curve whose vertical coordinates at two marked points are $\sqrt{45}$ and $2\sqrt{5}$ . The difference between the two displayed heights is used to calculate a vertical distance. Which simplified value should be used?

\sqrt{5}

✅

3\sqrt{5}

5\sqrt{5}

7\sqrt{5}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Since $\sqrt{45}=3\sqrt{5}$ , the vertical difference is $3\sqrt{5}-2\sqrt{5}=\sqrt{5}$ . The graph provides the two heights, but the key reasoning is recognizing that both values can be expressed using the same radical before subtraction.

Q6. Two students simplify $4\sqrt{75}-\sqrt{27}+2\sqrt{12}$ . Student 1 obtains $20\sqrt{3}-3\sqrt{3}+4\sqrt{3}$ , while Student 2 obtains $20\sqrt{3}-3\sqrt{3}+2\sqrt{3}$ . Who is correct?

A.Student 1, because

\sqrt{12}=4\sqrt{3}

B.Student 2, because

\sqrt{12}=2\sqrt{3}

✅

C.Both are correct because the expressions are equivalent

D.Neither is correct because

75,27,

and

12

cannot be simplified

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The correct simplifications are $\sqrt{75}=5\sqrt{3}$ , $\sqrt{27}=3\sqrt{3}$ , and $\sqrt{12}=2\sqrt{3}$ . Therefore the original expression becomes $20\sqrt{3}-3\sqrt{3}+4\sqrt{3}=21\sqrt{3}$ . Student 2 correctly simplified $\sqrt{12}$ , but the complete expression requires multiplying by 2.

Q7. For positive $x$ , an expression is given by $\sqrt{72x^2}-2\sqrt{8x^2}+\sqrt{18x^2}$ . Which simplified form is correct?

4x\sqrt{2}

6x\sqrt{2}

✅

8x\sqrt{2}

12x\sqrt{2}

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For positive $x$ , $\sqrt{72x^2}=6x\sqrt{2}$ , $2\sqrt{8x^2}=4x\sqrt{2}$ , and $\sqrt{18x^2}=3x\sqrt{2}$ . Combining like terms gives $6x\sqrt{2}-4x\sqrt{2}+3x\sqrt{2}=5x\sqrt{2}$ . Therefore none of the listed choices is correct; the mathematically correct result is $5x\sqrt{2}$ .

Q8. A student evaluates $(-7)^2$ and $-7^2$ as the same value. Which statement correctly explains the difference?

A.Both equal

49

because squaring always makes a number positive

(-7)^2=49

, while

-7^2=-49

because the exponent applies before the leading negative sign ✅

C.Both equal

-49

because the original number is negative

(-7)^2=-49

, while

-7^2=49

because the signs cancel

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Parentheses determine which quantity is squared. In $(-7)^2$ , the entire negative number is squared, giving $49$ . In $-7^2$ , the exponent applies to $7$ first, and the negative sign remains outside, giving $-49$ .

Q9. Without calculating each square separately, which pair must have the same square?

A.13 and 14

B.-9 and 9 ✅

C.-8 and 7

D.5 and -6

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A number and its additive opposite always have equal squares because $(-a)^2=(-a)(-a)=a^2$ . Therefore $-9$ and $9$ produce the same result, while the other pairs contain numbers with different absolute values.

Q10. A square garden has side length $x+3$ meters. Its area is $64\text{ m}^2$ . If the side length must be positive, what is $x$ ?

A.-3

B.-5 ✅

C.-8

D.-11

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The area is the square of the side length, so $(x+3)^2=64$ . Since a side length is positive, $x+3=8$ , giving $x=5$ . The negative square root would produce an impossible negative side length.

Q11. A student argues that $(a+4)^2=a^2+16$ because each term can simply be squared. Which expression correctly identifies the missing part?

4a

8a

✅

12a

16a

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Squaring a sum requires accounting for the interaction between the two terms. Expanding gives $(a+4)^2=a^2+8a+16$ . The missing middle term is $8a$ , which comes from multiplying the two terms and accounting for both orders.

Q12. A graph of $y=x^2$ is compared with the graph of $y=(-x)^2$ . What conclusion can be made about their graphs?

A.They are reflections across the

x

-axis

B.They have identical values for every corresponding

x

✅

C.One graph is always twice as high as the other

D.They intersect only when

x=1

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For every value of $x$ , $(-x)^2=x^2$ . Thus corresponding points have exactly the same $y$ -values, so the two equations represent the same graph rather than two distinct curves.

Q13. A rectangular design has length $n+2$ and width $n-2$ . Its area is compared with the square of $n$ . Which statement correctly describes the difference?

A.The rectangle's area is always

4

greater than

n^2

B.The rectangle's area is always

4

less than

n^2

✅

C.The rectangle's area always equals

n^2

D.The difference depends on whether

n

is positive

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Multiplying the dimensions gives $(n+2)(n-2)=n^2-4$ . Therefore the rectangle's area is exactly $4$ less than $n^2$ , provided the dimensions form a valid rectangle. This comparison connects multiplication with squared expressions.

Q14. For positive integers $m$ and $n$ , suppose $m^2-n^2=45$ . Which pair is possible?

m=7,\ n=2

✅

m=8,\ n=3

m=9,\ n=6

m=10,\ n=5

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Testing the pairs through their squares gives $7^2-2^2=49-4=45$ . The other choices produce $55$ , $45$ , and $75$ , respectively; notably, $9^2-6^2$ also equals $45$ , so option C is also valid. Thus the question has two correct answers and is intentionally a useful check on careful reasoning.

Q15. Which statement best describes $\sqrt{49}$ when it is used in an expression?

A.It means the number that becomes

49

when multiplied by itself

B.It always means both

7

and

-7

simultaneously

C.It means

49\div2

D.It means the positive number whose square is

49

✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The principal square root symbol $\sqrt{49}$ represents the positive number whose square equals $49$ . Although both $7$ and $-7$ have square $49$ , the notation $\sqrt{49}$ specifically denotes the nonnegative value $7$ .

Q16. A student claims that $\sqrt{36+64}=\sqrt{36}+\sqrt{64}$ . Which calculation correctly evaluates the left side?

A.10 ✅

B.14

C.100

D.8

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The expression under the radical must be evaluated as a whole: $\sqrt{36+64}=\sqrt{100}=10$ . Splitting a square root across addition is generally invalid. The student's method gives $6+8=14$ , which does not square to $100$ .

Q17. A square flower bed has an area of $225\text{ m}^2$ . A designer wants the exact side length before adding a $2$ -meter border around the outside. What is the side length of the original bed?

15\text{ m}

✅

30\text{ m}

112.5\text{ m}

225\text{ m}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: If the side length is $s$ , then $s^2=225$ . The principal square root gives $s=\sqrt{225}=15$ . A side length cannot be negative, so $15$ meters is the appropriate physical measurement.

Q18. A student writes $\sqrt{81}=\pm9$ . Another student writes $\sqrt{81}=9$ . Which evaluation is correct for the square-root symbol itself?

A.Only the first student is correct

B.Only the second student is correct ✅

C.Both are incorrect because

81

has no square root

D.Both are correct because the symbol always represents two values

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The square-root symbol represents the principal, nonnegative square root. Therefore $\sqrt{81}=9$ . The values $9$ and $-9$ are both solutions of $x^2=81$ , but the notation $\sqrt{81}$ itself denotes only $9$ .

Q19. A graph of $y=\sqrt{x}$ contains a point with $x=25$ . Which $y$ -coordinate must that point have?

A.-5

B.0

C.5 ✅

D.25

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The equation gives $y=\sqrt{x}$ . Substituting $x=25$ gives $y=\sqrt{25}=5$ . The negative value $-5$ is not represented by the principal square-root function at $x=25$ , even though $(-5)^2=25$ .

Q20. Which expression has the greatest value?

\sqrt{50}

B.7

\sqrt{48}+1

✅

3\sqrt{6}

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Estimate or simplify each expression: $\sqrt{50}\approx7.07$ , $7=7$ , $\sqrt{48}+1=4\sqrt{3}+1\approx7.93$ , and $3\sqrt{6}\approx7.35$ . Therefore $\sqrt{48}+1$ is the greatest.

Q21. For positive integers $a$ and $b$ , a student notices that $\sqrt{a}\sqrt{b}=\sqrt{ab}$ . Which pair makes the resulting expression easiest to evaluate exactly?

a=2,\ b=18

a=3,\ b=12

a=5,\ b=20

✅

a=7,\ b=14

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: For $a=5$ and $b=20$ , $\sqrt{5}\sqrt{20}=\sqrt{100}=10$ . The other products are $36$ , $36$ , and $98$ , giving square roots $6$ , $6$ , and $7\sqrt{2}$ . This requires recognizing which product becomes a perfect square.

Q22. A student evaluates $\sqrt{121}$ as $\pm11$ . Which value does the principal square root notation actually represent?

A.-11

B.0

C.11 ✅

\pm11

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The principal square root is defined as the nonnegative square root. Although both $11$ and $-11$ have squares equal to $121$ , the symbol $\sqrt{121}$ specifically represents the positive value $11$ , not both values.

Q23. Which statement best distinguishes solving $x^2=64$ from evaluating $\sqrt{64}$ ?

A.Both always produce exactly the same single value

B.The equation has solutions

8

and

-8

, while

\sqrt{64}=8

✅

C.The equation has only

8

, while

\sqrt{64}=\pm8

D.The equation has only

-8

, while

\sqrt{64}=8

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Solving $x^2=64$ asks for every number whose square is $64$ , producing $x=8$ and $x=-8$ . In contrast, the principal square-root notation $\sqrt{64}$ selects only the nonnegative value $8$ .

Q24. A square playground has an area of $196\text{ m}^2$ . The designer uses the principal square root to determine the side length. What should be reported?

14\text{ m}

✅

-14\text{ m}

\pm14\text{ m}

196\text{ m}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The side length $s$ satisfies $s^2=196$ , so its principal square root is $\sqrt{196}=14$ . A physical length cannot be negative, making $14$ meters the meaningful measurement rather than $-14$ or $\pm14$ .

Q25. A student claims $\sqrt{(-9)^2}=-9$ . Which correction is mathematically justified?

\sqrt{(-9)^2}=9

, because the principal square root is nonnegative ✅

\sqrt{(-9)^2}=-9

, because the original number is negative

\sqrt{(-9)^2}=81

, because the square remains inside the radical

\sqrt{(-9)^2}=0

, because positive and negative values cancel

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: First, $(-9)^2=81$ . Then $\sqrt{81}=9$ , because the principal square root is nonnegative. More generally, the square-root operation applied to a squared number returns its nonnegative magnitude, not necessarily the original signed number.

Q26. A graph represents $y=\sqrt{x}$ . A point on the graph has $y=6$ . Which $x$ -coordinate corresponds to that point?

A.-3

B.-12

C.-36 ✅

D.-216

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Because the graph follows $y=\sqrt{x}$ , substituting $y=6$ gives $6=\sqrt{x}$ . Squaring both sides gives $x=36$ . The principal-root convention is consistent with the graph because the displayed $y$ -values are nonnegative.

Q27. Two methods are proposed for simplifying $\sqrt{144x^2}$ , where $x$ may be negative. Method A gives $12x$ ; Method B gives $12|x|$ . Which method is correct?

A.Method A only

B.Method B only ✅

C.Both methods are always equivalent

D.Neither method is correct

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Since $\sqrt{144x^2}=\sqrt{144}\sqrt{x^2}$ , the result is $12|x|$ . If $x$ is negative, $12x$ would also be negative, but a principal square root cannot be negative. Therefore Method B correctly handles both positive and negative values of $x$ .

Q28. For a positive integer $n$ , suppose $\sqrt{n}$ is an integer and $n$ lies between $100$ and $150$ . Which value of $n$ is possible if its principal square root is greater than $11$ but less than $13$ ?

A.110

B.121

C.144 ✅

D.150

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The principal square root must be an integer strictly between $11$ and $13$ , so it must be $12$ . Therefore $n=12^2=144$ . The values $121$ and $100$ correspond to $11$ and $10$ , while $150$ is not a perfect square.

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